Measurement device and measurement method

ABSTRACT

A measurement device applies a pulse current or a current of a plurality of frequencies to a laminated body having a plurality of layers having different electrical conductivities. The device acquires in-plane distribution information indicating distribution of a magnetic field of a first plane outside the laminated body. A processor generates three-dimensional magnetic field distribution information indicating a three-dimensional distribution of magnetic field on an outside of the laminated body based on the in-plane distribution information, and generates information on an inside of the laminated body by processing the three-dimensional magnetic field distribution information on the outside of the laminated body. The in-plane distribution information includes information indicating response characteristics to a change in the current applied by the current applying unit. In addition, the three-dimensional magnetic field distribution information on the outside of the laminated body includes frequency characteristics of the magnetic field.

TECHNICAL FIELD

The present invention relates to a measurement device and a measurementmethod.

BACKGROUND ART

In recent years, technologies for inspecting a defect of a battery, anelectronic element, and the like have been developed. For example,nondestructively specifying a defect occurring inside a lithium batteryand the like is also important for specifying a cause of failure.

For example, in Patent Document 1, matters that a state of a battery isnondestructively inspected using infrared rays, visible rays, X-rays,gamma rays, ultrasonic waves, and the like and is evaluated incombination with a comparison value are described.

For example, in Patent Document 2, a method of deriving internalelectrical conductivity distribution from the result of measuring amagnetic field outside the battery.

RELATED DOCUMENTS Patent Documents

[Patent Document 1] PCT Japanese Translation Patent Publication No.2012-524385

[Patent Document 2] Pamphlet of International Publication No. WO2015/136931

SUMMARY OF THE INVENTION Technical Problem

However, with the method of Patent Document 1, it is difficult toanalyze the inspection result in a state where there is no comparisonvalue.

Also, with the method of Patent Document 2, it is difficult to evaluatethe inside of a structure in which many layers are laminated.

An object of the present invention is to provide a measurement deviceand a measurement method for obtaining information on an inside of alaminated body by using a magnetic field.

Solution to Problem

According to the present invention, there is provided a measurementdevice including a current applying unit which applies a pulse currentor a current of a plurality of frequencies to a laminated body having astructure in which a plurality of layers having different electricalconductivities from each other are laminated, an acquisition unit whichacquires in-plane distribution information including at leastinformation indicating distribution of a magnetic field of a first planeoutside the laminated body, and a processing unit which generatesthree-dimensional magnetic field distribution information indicating athree-dimensional distribution of magnetic field on an outside of thelaminated body based on the in-plane distribution information andgenerates information on an inside of the laminated body by processingthe three-dimensional magnetic field distribution information on theoutside of the laminated body, in which the in-plane distributioninformation includes frequency dependent complex data which isinformation indicating response characteristics to a change in currentapplied by the current applying unit, the three-dimensional magneticfield distribution information on the outside of the laminated bodyincludes frequency characteristics of a magnetic field, the processingunit specifies a position including depth information of a defect insidethe laminated body based on the three-dimensional magnetic fielddistribution information on the outside of the laminated body, aboundary condition between the magnetic fields inside and outside thelaminated body, and an equation relating to the magnetic field insidethe laminated body, and the equation is an averaged and continuousdiffusion type partial differential equation of the laminated body.

According to the present invention, there is provided a measurementmethod including applying a pulse current or a current of a plurality offrequencies to a laminated body having a structure in which a pluralityof layers having different electrical conductivities from each other arelaminated, acquiring in-plane distribution information including atleast information indicating distribution of a magnetic field of a firstplane outside the laminated body, generating three-dimensional magneticfield distribution information indicating a three-dimensionaldistribution of magnetic field on an outside of the laminated body basedon the in-plane distribution information and generating information onan inside of the laminated body by processing the three-dimensionalmagnetic field distribution information on the outside of the laminatedbody, in which the in-plane distribution information includes frequencydependent complex data which is information indicating responsecharacteristics to a change in current applied to the laminated body,the three-dimensional magnetic field distribution information on theoutside of the laminated body includes frequency characteristics of amagnetic field, in the generating of the information on the inside ofthe laminated body, a position including depth information of a defectinside the laminated body is specified based on the three-dimensionalmagnetic field distribution information on the outside of the laminatedbody, a boundary condition between the magnetic fields inside andoutside the laminated body, and an equation relating to the magneticfield inside the laminated body, and the equation is an averaged andcontinuous diffusion type partial differential equation of the laminatedbody.

Advantageous Effects of Invention

According to the present invention, it is possible to obtain informationon an inside of the laminated body by using a magnetic field.

BRIEF DESCRIPTION OF THE DRAWINGS

The object described above and other objects, features and advantageswill become more apparent from the following description of thepreferred embodiments and the accompanying drawings.

FIG. 1 is a block diagram illustrating a functional configuration of ameasurement device according to a first embodiment.

FIG. 2 is a cross-sectional view illustrating an example of a structureof a laminated body according to the first embodiment.

FIG. 3 is a diagram illustrating a configuration example of themeasurement device according to the first embodiment.

FIG. 4 is a block diagram illustrating a functional configuration of aprocessing unit according to the first embodiment.

FIG. 5 is a block diagram illustrating a functional configuration of aprocessing unit according to a third embodiment.

FIG. 6 is a view illustrating a structure of a battery having amultilayer structure.

FIG. 7 is a view illustrating a boundary between a layer and a layer.

FIG. 8 is a diagram illustrating definitions of variables in thelaminated body.

FIG. 9 is a view illustrating a boundary between a layer and a layer.

FIGS. 10A to 10D are graphs illustrating examples of a relationshipbetween z and Φ obtained in a fourth embodiment.

DESCRIPTION OF EMBODIMENTS

Hereinafter, embodiments of the present invention will be described withreference to the drawings. In all the drawings, similar constitutionalelements are denoted by the same reference numerals, and descriptionthereof is omitted as appropriate.

In the following description, a current applying unit 120, anacquisition unit 140, a processing unit 160, a display unit 180, astoring unit 190, an outside three-dimensional distribution generationunit 162, an inside three-dimensional distribution generation unit 164,and a defect specifying unit 166 of a measurement device 10 illustratenot hardware units but blocks of functional units. The current applyingunit 120, the acquisition unit 140, the processing unit 160, the displayunit 180, the storing unit 190, the outside three-dimensionaldistribution generation unit 162, the inside three-dimensionaldistribution generation unit 164, and the defect specifying unit 166 ofthe measurement device 10 are realized by any combination of hardwareand software, mainly on a CPU, a memory, a program loaded into thememory realizing constituent elements in these figures, a storage mediumsuch as a hard disk storing the program, and a network connectioninterface of any computer. There are various modifications to theimplementation method and apparatus.

In the following description, a bold character indicating a vector in amathematical expression may be represented by characters that are notbold in the sentence.

First Embodiment

FIG. 1 is a block diagram illustrating a functional configuration of themeasurement device 10 according to a first embodiment. The measurementdevice 10 includes the current applying unit 120, the acquisition unit140, and the processing unit 160. The current applying unit 120 appliesa pulse current or a current having a plurality of frequencies to alaminated body having a structure in which a plurality of layers havingdifferent electrical conductivities from each other are laminated. Theacquisition unit 140 acquires in-plane distribution informationincluding at least information indicating a distribution of a magneticfield of a first plane outside the laminated body. The processing unit160 generates three-dimensional magnetic field distribution informationindicating a three-dimensional distribution of magnetic field on anoutside of the laminated body based on the in-plane distributioninformation, and generates information on the inside of the laminatedbody by processing the three-dimensional magnetic field distributioninformation on the outside of the laminated body. Here, the in-planedistribution information includes information indicating a responsecharacteristic to a change in current applied by the current applyingunit 120. The three-dimensional magnetic field distribution informationon the outside of the laminated body includes a frequency characteristicof the magnetic field. In the following, description will be made indetail.

The measurement device 10 according to the present embodiment furtherincludes a display unit 180. The display unit 180 displays informationon the inside of the laminated body generated by the processing unit160. However, the measurement device 10 may not include the display unit180, and the processing unit 160 may output data indicating theinformation on the inside of the laminated body to another device, forexample.

The measurement device 10 according to the present embodiment furtherincludes the storing unit 190. However, the storing unit 190 may be anexternal memory or the like separately provided from the measurementdevice 10. The processing unit 160 can read out information and the likeillustrating mathematical expressions to be described later from thestoring unit 190 and use the information and the like for processing.

FIG. 2 is a cross-sectional view illustrating an example of a structureof a laminated body 20 according to the present embodiment. In theexample of this figure, the laminated body 20 is a battery such as alithium battery, for example. In a case where the laminated body 20 is,for example, a battery, the laminated body 20 has a structure in which afirst layer 210 and a second layer 220 having different electricalconductivities from each other are alternately laminated. Both the firstlayer 210 and the second layer 220 are planar. For example, the firstlayer 210 is a conductor layer and the second layer 220 is anelectrolyte layer. The first layer 210 may be composed of a singlelayer, or an electrode composed of a plurality of layers havingdifferent materials from each other. In addition, the first layer 210includes a positive electrode 210 a and a negative electrode 210 bcontaining different materials from each other. The electricalconductivities of the positive electrode 210 a and the negativeelectrode 210 b need not be perfectly identical, but the electrodes areapproximately the same, and each of the electrodes functions as anelectrode. The positive electrode 210 a and the negative electrode 210 bare alternately disposed. That is, each of the second layers 220 issandwiched between the positive electrode 210 a and the negativeelectrode 210 b. In the example of this figure, the laminated body 20includes a plurality of positive electrodes 210 a and a plurality ofnegative electrodes 210 b. The plurality of positive electrodes 210 aare connected to a first electrode 203 a (see FIG. 3) to be describedlater, and the plurality of negative electrodes 210 b are connected to asecond electrode 203 b (see FIG. 3) to be described later. The laminatedbody 20 may be further covered with a protective layer or the like.

Here, when it is assumed that two directions parallel to the first layer210 and orthogonal to each other are the x-direction and the y-directionand a direction perpendicular to the x-direction and y-direction is thez-direction, a first plane 201 is a plane parallel to the xy-plane. Inthe example of this figure, one surface of the laminated body 20 is setto z=0, and the direction toward the inside of the laminated body 20 isset as the positive direction of the z-axis.

In the present embodiment, the in-plane distribution informationincludes information indicating the distribution of the magnetic fieldon the first plane 201 and information indicating the distribution ofthe magnetic field on the second plane 202. Each of the first plane 201and the second plane 202 is a plane outside the laminated body 20, thesecond plane 202 is parallel to the first plane 201 and the z-coordinatethereof is different from the z-coordinate of the first plane 201. Amagnitude relation between the z-coordinates of the first plane 201 andthe second plane 202 is not particularly limited. In the example of thisfigure, the z-coordinates of the first plane 201 and the second plane202 are both negative.

FIG. 3 is a diagram illustrating a configuration example of themeasurement device 10 according to the present embodiment. Eachconstitutional element of the measurement device 10 will be describedwith reference to FIGS. 1 to 3.

The current applying unit 120 applies a pulsed current (pulse current)or a current having a plurality of frequencies to a laminated bodyhaving a structure in which a plurality of layers having differentelectrical conductivities from each other are laminated. The currentapplying unit 120 is constituted with, for example, a current generator340, a current stabilization power supply 342, and an informationprocessing device 380. The information processing device 380 is, forexample, a computer including a CPU, a memory, a program loaded into amemory, a storage medium such as a hard disk that stores the program, anetwork connection interface, and the like. The current stabilizationpower supply 342 supplies power to the current generator 340. Thecurrent generator 340 outputs a pulse current based on a control signalfrom the information processing device 380. Alternatively, the currentgenerator 340 outputs an alternating current based on a control signalfrom the information processing device 380 and sweeps the frequency ofthe current. The current generator 340 is electrically connected to oneof the layers constituting the laminated body 20. In the example of FIG.3, the laminated body 20 has the structure illustrated in FIG. 2, thecurrent generator 340 is connected to the first electrode 203 a and thesecond electrode 203 b of the laminated body 20, and a current flowsbetween the first electrode 203 a and the second electrode 203 b. Thestructure of the laminated body 20 to be measured is not limited to thestructure illustrated in FIG. 2. For example, it may be a sample inwhich all the first layers 210 are not electrically connected to eachother. In that case, the current from the current applying unit 120 maybe applied to at least one layer of the laminated body. The currentgenerator 340 outputs a reference signal synchronized with the currentapplied to the laminated body 20.

The laminated body 20 is disposed on a sample stage 35. The sample stage35 includes an XYZ drive stage 350 and a e drive stage 352, and canchange a relative position of the laminated body 20 with respect to amagnetic sensor 312 described later on the basis of a control signalfrom the information processing device 380. Specifically, a signalindicating the coordinates to be taken by each stage or a signalindicating a drive amount is output from the information processingdevice 380 to a feedback control circuit 362 and a feedback controlcircuit 372. Then, in the feedback control circuit 362 and the feedbackcontrol circuit 372, signals indicating the voltages to be applied tothe XYZ drive stage 350 and the θ drive stage 352 are generated andoutput, respectively. Specifically, the feedback control circuit 362 andthe feedback control circuit 372 perform feedback control based onoutputs of position sensors provided in the XYZ drive stage 350 and theθ drive stage 352, respectively, to generate a signal so as to allowdriving to a desired position or a desired amount of driving. A highvoltage power supply 360 and a high voltage power supply 370 amplify thesignals from the feedback control circuit 362 and the feedback controlcircuit 372, respectively, and input the amplified signals to the XYZdrive stage 350 and the θ drive stage 352, respectively. The XYZ drivestage 350 and the θ drive stage 352 are driven in accordance with inputsignals from the high voltage power supply 360 and the high voltagepower supply 370, respectively.

In the measurement, the sample stage 35 is driven such that the magneticsensor 312 scans in the xy-plane. The magnetic sensor 312 is attached tothe sensor holding unit 310. A distance sensor such as an electrostaticcapacitance sensor is attached to a sensor holding unit 310 and canmeasure the distance between the magnetic sensor 312 and the laminatedbody 20 or the distance between the magnetic sensor 312 and the samplestage 35. Then, based on the measured distance, z driving of the XYZdrive stage 350 is feedback controlled so that it is possible to scanover the laminated body 20 at a predetermined z-coordinate. By doing so,the magnetic sensor 312 can measure the magnetic field at each point inthe xy-plane at the desired z-coordinate.

While the magnetic sensor 312 scans in the first plane 201 and thesecond plane 202, the current applying unit 120 applies pulses one byone at each lattice point represented by the xy-coordinates, forexample. Alternatively, the current applying unit 120 sweeps the currentfrequency at each lattice point in a predetermined range.

The acquisition unit 140 acquires in-plane distribution informationincluding at least information indicating the distribution of themagnetic field of the first plane 201 outside the laminated body 20. Theacquisition unit 140 is constituted with, for example, a magnetic sensor312, a preamplifier 322, a phase detection circuit 320, and theinformation processing device 380.

The magnetic sensor 312 is a magnetic sensor such as a tunnel magnetoresistance (TMR) element. The magnetic field outside the laminated body20 can be measured by the magnetic sensor 312. Specifically, themagnetic sensor 312 can measure the x-component, the y-component, thez-component of the magnetic field at a plurality of points in the firstplane 201 and the second plane 202, and their time change. Theacquisition unit 140 may acquire the in-plane distribution informationfrom another measurement device.

Only one magnetic sensor is attached to the magnetic sensor 312, and themagnetic field at one x-coordinate may be measured every time one linescanning is performed, or a plurality of magnetic sensors may bedisposed in an array and magnetic field information of a plurality ofx-coordinates may be acquired by performing one line scanning. Thex-component, the y-component, and the z-component of the magnetic fieldmay be measured simultaneously by a single scan, or may be measured byscanning twice or three times by changing the component to be measured.

The in-plane distribution information includes magnetic fieldinformation on a magnetic field at least for each of a plurality ofpoints in the first plane 201. Here, the magnetic field informationincludes information indicating a response characteristic of thex-component, the y-component, and the z-component of the magnetic field.As described above, the response characteristic is the responsecharacteristic of the magnetic field with respect to the change in thecurrent applied by the current applying unit 120. That is, the magneticfield information can be represented by a complex magnetic field vector.

Here, FIG. 3 illustrates an example of a case where the current applyingunit 120 applies the current having a plurality of frequencies to thelaminated body 20, that is, a case where the frequency of the current isswept. An output signal from the magnetic sensor 312 is input to thepreamplifier 322 and amplified, and then input to the phase detectioncircuit 320 as a measurement signal. Based on the signal input from thepreamplifier 322 and the reference signal from the current generator340, the phase detection circuit 320 associates time change of themeasured magnetic field with time change of the current applied to thelaminated body 20. Specifically, the phase detection circuit 320 outputsa signal indicating an amplitude of the input measurement signal and asignal indicating a phase difference between the measurement signal andthe reference signal for each component of the x, y, and z-components.In the information processing device 380, magnetic field information ateach xy-coordinate represented by a complex magnetic field vector isobtained on the basis of signals indicating the amplitude and phasedifference. In this way, the acquisition unit 140 acquires the in-planedistribution information.

On the other hand, in a case where the current applying unit 120 appliesthe pulse current to the laminated body 20, the phase detection circuit320 can be omitted. In this case, the measurement signal output from thepreamplifier 322 and the reference signal from the current generator 340are input to the information processing device 380. Then, theinformation processing device 380 generates information indicating apulse response in association with the measured magnetic field so thatthe rising time point of the applied pulse current is set to t=0. In theinformation processing device 380, information indicating the pulseresponse is subjected to inverse Fourier transform, thereby obtainingmagnetic field information at each xy-coordinate represented by acomplex magnetic field vector. In this way, the acquisition unit 140acquires the in-plane distribution information.

The magnetic field information input to the information processingdevice 380 is subjected to further processing such as resolutionenhancement, and the processing unit 160 may generate thethree-dimensional magnetic field distribution information on the outsideof the laminated body 20 by using the in-plane distribution informationafter processing as described later.

The processing unit 160 generates three-dimensional magnetic fielddistribution information indicating the three-dimensional distributionof magnetic field on the outside of the laminated body 20, based on thein-plane distribution information. The processing unit 160 isconstituted by the information processing device 380.

FIG. 4 is a block diagram illustrating a functional configuration of theprocessing unit 160 according to the first embodiment. In the presentembodiment, the processing unit 160 includes the outsidethree-dimensional distribution generation unit 162 and the insidethree-dimensional distribution generation unit 164. The outsidethree-dimensional distribution generation unit 162 processes thein-plane distribution information acquired by the acquisition unit 140to generate three-dimensional magnetic field distribution information onthe outside of the laminated body 20. The inside three-dimensionaldistribution generation unit 164 processes the three-dimensionalmagnetic field distribution information on the outside of the laminatedbody 20 generated by the outside three-dimensional distributiongeneration unit 162 to generate three-dimensional magnetic fielddistribution information on the inside of the laminated body 20 asinformation on the inside of the laminated body 20.

First, processing contents of the outside three-dimensional distributiongeneration unit 162 will be described below. The three-dimensionalmagnetic field distribution information on the outside of the laminatedbody 20 is represented by the following expression (1), ands isrepresented by the following expression (2). Here, k_(x) is a wavenumber in the x-direction of the magnetic field, k_(y) is the wavenumber in the y-direction of the magnetic field, z is a z-coordinate, ωis a frequency, σ is electrical conductivity, μ is magneticpermeability, and the following expression (3) is a complex magneticfield vector. In the present embodiment, an outer space of the laminatedbody 20 is air, σ is the electrical conductivity of air, and μ is themagnetic permeability of air. Then, the outside three-dimensionaldistribution generation unit 162 obtains vectors a(k_(x), k_(y), ω) andb(k_(x), k_(y), ω) in the equation (1) using information indicating thedistribution of the magnetic field of the first plane 201 and theinformation indicating the distribution of the magnetic field of thesecond plane 202. Each mathematical expression will be described laterin detail.

$\begin{matrix}{\mspace{79mu}{{\overset{\sim}{H}\left( {k_{x},k_{y},z,\omega} \right)} = {{{a\left( {k_{x},k_{y},\omega} \right)}e^{sz}} + {{b\left( {k_{x},k_{y},\omega} \right)}e^{- {sz}}}}}} & (1) \\{s = {\frac{\sqrt{k_{x}^{2} + k_{y}^{2} + \sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\omega^{2}\sigma^{2}\mu^{2}}}}}{\sqrt{2}} + {i\frac{\sqrt{{- k_{x}^{2}} - k_{y}^{2} + \sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\omega^{2}\sigma^{2}\mu^{2}}}}}{\sqrt{2}}}}} & (2) \\{\mspace{79mu}{\overset{\sim}{H}\left( {k_{x},k_{y},z,\omega} \right)}} & (3)\end{matrix}$

That is, the outside three-dimensional distribution generation unit 162can process as follows. First, the outside three-dimensionaldistribution generation unit 162 acquires in-plane distributioninformation. Here, the in-plane distribution information includesinformation indicating the complex magnetic field vector at eachcoordinate as described above. It is assumed that the z-coordinate ofthe first plane 201 is z₁, the z-coordinate of the second plane 202 isz₂, information indicating the distribution of the magnetic field of thefirst plane 201 is,H ₁(x,y,z ₁,ω)  (4)and information indicating the distribution of the magnetic field of thesecond plane 202 isH ₂(x,y,z ₂,ω)  (5)

Then, the outside three-dimensional distribution generation unit 162obtains{tilde over (H)} ₁(k _(x) ,k _(y) ,z ₁,ω)  (6)by performing Fourier transformation on the information indicating thedistribution of the magnetic field of the first plane 201 with respectto x and y, and obtains{tilde over (H)} ₂(k _(x) ,k _(y) ,z ₂,ω)  (7)by performing Fourier transformed on information indicating thedistribution of the magnetic field of the second plane 202 with respectto x and y.

Then, a three-dimensional distribution of the complex magnetic fieldvector represented by the expression (1) is generated as thethree-dimensional magnetic field distribution information on the outsideof the laminated body 20. Here, a and b in the expression (1) arevectors which satisfy the following expressions (8) and (9).{tilde over (H)} ₁(k _(x) ,k _(y) ,z ₁,ω)=a(k _(x) ,k _(y),ω)e ^(sz) ¹+b(k _(x) ,k _(y),ω)e ^(−sz) ¹   (8){tilde over (H)} ₂(k _(x) ,k _(y) ,z ₂,ω)=a(k _(x) ,k _(y),ω)e ^(sz) ²+b(k _(x) ,k _(y),ω)e ^(−sz) ²   (9)

In this manner, the three-dimensional magnetic field distributioninformation on the outside of the laminated body 20 is generated fromthe in-plane distribution information including information indicatingthe response characteristic to the change in the current applied by thecurrent applying unit 120. The three-dimensional magnetic fielddistribution information on the outside of the laminated body 20includes the frequency characteristic of the magnetic field. Thethree-dimensional magnetic field distribution information on the outsideof the laminated body 20 is data represented by, for example, amathematical expression or a table.

Next, the inside three-dimensional distribution generation unit 164 ofthe processing unit 160 generates information on the inside of thelaminated body 20 by processing the three-dimensional magnetic fielddistribution information on the outside of the laminated body 20. In thepresent embodiment, the processing unit 160 generates thethree-dimensional magnetic field distribution information on the insideof the laminated body 20 based on the three-dimensional magnetic fielddistribution information on the outside of the laminated body 20.Processing performed by the inside three-dimensional distributiongeneration unit 164 will be described below.

The inside three-dimensional distribution generation unit 164 acquiresthe three-dimensional magnetic field distribution information on theoutside of the laminated body 20 from the outside three-dimensionaldistribution generation unit 162. Then, the inside three-dimensionaldistribution generation unit 164 generates the three-dimensionalmagnetic field distribution information on the inside of the laminatedbody 20 based on the three-dimensional magnetic field distributioninformation on the outside of the laminated body 20, the boundarycondition between the magnetic fields inside and outside the laminatedbody 20, and a plurality of relational expressions to be satisfied bythe magnetic field inside the laminated body 20.

Specifically, the boundary condition between the magnetic fields insideand outside the laminated body 20 is represented by the expression (10).

$\begin{matrix}{\mspace{79mu}{\left. {\overset{\sim}{H}}_{x} \right|_{inside} = {\left. {\overset{\sim}{H}}_{x} \middle| {}_{outside}\mspace{79mu}{\overset{\sim}{H}}_{y} \right|_{inside} = {\left. {\overset{\sim}{H}}_{y} \middle| {}_{outside}\mspace{79mu}{\overset{\sim}{H}}_{z} \right|_{inside} = {\left. {\overset{\sim}{H}}_{z} \middle| {}_{outside}{\sigma_{a}^{- 1}\left( {{{- {ik}_{y}}{\overset{\sim}{H}}_{z}} - \frac{\partial{\overset{\sim}{H}}_{y}}{\partial z}} \right)} \right|_{outside} = {\left. {\sigma_{c}^{- 1}\left( {{{- {ik}_{y}}{\overset{\sim}{H}}_{z}} - \frac{\partial{\overset{\sim}{H}}_{y}}{\partial z}} \right)} \middle| {}_{inside}\mspace{79mu}{\sigma_{a}^{- 1}\left( {{{- {ik}_{x}}{\overset{\sim}{H}}_{z}} - \frac{\partial{\overset{\sim}{H}}_{x}}{\partial z}} \right)} \right|_{outside} = {\left. {\sigma_{c}^{- 1}\left( {{{- {ik}_{x}}{\overset{\sim}{H}}_{z}} - \frac{\partial{\overset{\sim}{H}}_{x}}{\partial z}} \right)} \middle| {}_{inside}\mspace{79mu}\frac{\partial{\overset{\sim}{H}}_{z}}{\partial z} \right|_{outside} = \left. \frac{\partial{\overset{\sim}{H}}_{z}}{\partial z} \right|_{inside}}}}}}}} & (10)\end{matrix}$

Here,{tilde over (H)} _(x)is the x-component of the complex magnetic field vector,{tilde over (H)} _(y)is the y-component of the complex magnetic field vector,{tilde over (H)} _(z)is the z-component of the complex magnetic field vector, σ_(a) is theelectrical conductivity of outside the laminated body 20, and σ_(c) isthe electrical conductivity of the first layer 210. Here, the firstlayer 210 is the outermost layer on the first plane 201 side of thelaminated body 20.

Specifically, a plurality of relational expressions to be satisfied bythe magnetic field inside the laminated body 20 is expressed by theequation (11). Here, μ is magnetic permeability, t is time, H_(x) is acomponent in the x-direction of the magnetic field, H_(y) is a componentin the y-direction of the magnetic field, H_(z) is a component in thez-direction of the magnetic field, g_(c) is a thickness of the firstlayer 210, g_(e) is a thickness of the second layer 220, σ_(c) is theelectrical conductivity of the first layer 210, and σ_(e) is theelectrical conductivity of the second layer 220. Each mathematicalexpression will be described later in detail.

$\begin{matrix}{{{\mu\frac{\partial}{\partial t}H_{x}} = {{\left\{ {{\frac{\left( {{g_{e}\sigma_{c}} + {g_{c}\sigma_{e}}} \right)}{\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}} \right)} + {\frac{\left( {g_{c} + g_{e}} \right)}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)}\frac{\partial^{2}}{\partial z^{2}}}} \right\} H_{x}} + {\frac{g_{e}{g_{c}\left( {\sigma_{c} - \sigma_{e}} \right)}^{2}}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\frac{\partial^{2}}{{\partial x}{\partial z}}H_{z}}}}{{\mu\frac{\partial}{\partial t}H_{y}} = {{\left\{ {{\frac{\left( {{g_{e}\sigma_{c}} + {g_{c}\sigma_{e}}} \right)}{\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial}{\partial y^{2}}} \right)} + {\frac{\left( {g_{c} + g_{e}} \right)}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)}\frac{\partial^{2}}{\partial z^{2}}}} \right\} H_{y}} + {\frac{g_{e}{g_{c}\left( {\sigma_{c} - \sigma_{e}} \right)}^{2}}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\frac{\partial}{{\partial y}{\partial z}}H_{z}}}}\mspace{79mu}{{\mu\frac{\partial}{\partial t}H_{z}} = {\frac{\left( {g_{e} + g_{c}} \right)}{\left( {{g_{e}\sigma_{e}} + {g_{c}\sigma_{c}}} \right)}\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}}} \right)H_{z}}}} & (11)\end{matrix}$

In the boundary condition, the magnetic field outside the laminated body20 and its differential are derived from the expression (1) describedabove. As a solution to the expression (11), three-dimensional magneticfield distribution information on the inside of the laminated body 20 isobtained. The three-dimensional magnetic field distribution informationon the inside of the laminated body 20 is represented by H(x, y, z).Here, H(x, y, z) includes information indicating the x-component, they-component, and the z-component of the magnetic field of eachcoordinate. The three-dimensional magnetic field distributioninformation on the inside of the laminated body 20 is data representedby, for example, a mathematical expression or a table. Thethree-dimensional magnetic field distribution information on the insideof the laminated body 20 may not include the frequency characteristic.

The three-dimensional magnetic field distribution information on theinside of the laminated body 20 output from the inside three-dimensionaldistribution generation unit 164 is input to the display unit 180, forexample. The display unit 180 is constituted with, for example, theinformation processing device 380 and a monitor 390. In the display unit180, data for displaying an image of the three-dimensional magneticfield distribution information on the inside of the laminated body 20 isgenerated. Then, the distribution of the magnetic field inside thelaminated body 20 can be displayed as an image on the monitor 390.

In the following, a measurement method according to the presentembodiment will be described. This measurement method is realized by themeasurement device 10 as described above.

In the measurement method according to the present embodiment, a pulsecurrent or a current having a plurality of frequencies is applied to thelaminated body 20 having a structure in which a plurality of layershaving different electrical conductivities from each other are laminatedand in-plane distribution information including at least informationindicating the distribution of the magnetic field of the first plane 201outside the laminated body 20 is acquired. Then, based on the in-planedistribution information, three-dimensional magnetic field distributioninformation indicating the three-dimensional distribution of magneticfield on the outside of the laminated body 20 is generated. Furthermore,information on the inside of the laminated body 20 is generated byprocessing the three-dimensional magnetic field distribution informationon the outside of the laminated body 20. Here, the in-plane distributioninformation includes information indicating the response characteristicto a change in current applied to the laminated body 20 and thethree-dimensional magnetic field distribution information on the outsideof the laminated body 20 includes the frequency characteristic of themagnetic field. In the following, description will be made in detail.

First, while the current applying unit 120 applies a pulse current or acurrent having a plurality of frequencies to the laminated body 20, theacquisition unit 140 acquires the in-plane distribution informationincluding at least information indicating the distribution of themagnetic field of the first plane 201 outside the laminated body 20.Specifically, information indicating the response characteristic of themagnetic field with respect to the change in current is obtained byscanning the measurement positions of the magnetic field with respect tothe laminated body 20 and applying a pulse current at each measurementpoint or by sweeping the frequency of the current at each measurementpoint. In the present embodiment, measurement is similarly performed onthe second plane 202 to acquire the in-plane distribution information.

As described above, the processing unit 160 processes the in-planedistribution information including the information indicating thedistribution of the magnetic field of the first plane 201 and theinformation indicating the distribution of the magnetic field of thesecond plane 202 to thereby generate the three-dimensional magneticfield distribution information on the outside of the laminated body 20.Furthermore, the processing unit 160 processes the three-dimensionalmagnetic field distribution information on the outside of the laminatedbody 20 to thereby generate the three-dimensional magnetic fielddistribution information on the inside of the laminated body 20 asinformation on the inside of the laminated body 20.

The measurement device 10 may be a measurement device unit that can beattached to an existing device.

Next, operations and effects of the present embodiment will bedescribed. According to the measurement device 10 of the presentembodiment, it is possible to obtain information on the inside of thelaminated body 20 such as a battery. Furthermore, it provides usefulinformation for non-destructive failure analysis and the like.

In the present embodiment, it is possible to know the three-dimensionalmagnetic field distribution inside the laminated body 20 such as abattery. Therefore, it is possible to specifically know the situationinside the laminated body 20.

In the following, description will be made on analysis of the magneticfield inside and outside the laminated body 20. In the followingdescription, the laminated body 20 is described as a battery, but thelaminated body 20 is not limited to the battery.

<I Overview>

It is difficult to inspect inside of a battery having a multilayerstructure using a magnetic field with a method in which a DC magneticfield is used. It is necessary to measure a new physical quantity inorder to obtain information in the z-direction which is the directionperpendicular to a battery surface. The frequency characteristic isconsidered as the new physical quantity. Since the inside of the batteryis a conductor, an internal magnetic field satisfies a diffusion typepartial differential equation. Accordingly, information obtained isconsiderably restricted as compared with the case where the waveequation is satisfied.

In the following, electrodes and electrolytes inside the battery areaveraged and made continuous, and a diffusion equation is derived. Next,the boundary condition on the battery surface of this equation isderived. The boundary condition is obtained by measuring the frequencydependent complex data of the magnetic field at a place away from thesurface. For the diffusion equation which is made continuous, an inverseproblem can be analyzed by using the boundary condition. By setting timet=0, a final solution to the inverse problem can be obtained. Theseprocesses are similar to inverse analysis using the Laplace's equationin the case of the conventional static magnetic field, which areextension to a natural time dependent system. As a result, athree-dimensional image can be obtained.

In the following “II Assumption of 2D magnetic field (B_(x), B_(y), 0)”,it is assumed that the magnetic field is two dimensional (x, y plane).In the following “III Basic Theory Assumption of 3D magnetic field(B_(x), B_(y), B_(z))”, the same result as in the case of two dimensionsis obtained assuming that the magnetic field is three-dimensional. Thiscorresponds to a case where the z-component of the magnetic fieldincluded in the in-plane distribution information described in the firstembodiment is measured and used for processing.

<II Assumption of 2D Magnetic Field (B_(x), B_(y), 0)>

<II-1 Averaged Basic Equation>

A practical battery has a multilayer structure. Since the existingmagnetic imaging technology is a two-dimensional (x-y plane) imagingtechnology, it was difficult to find out which layer of the multilayerthere is an abnormal portion. Image reconstruction has been performedusing measured data with DC or at a fixed frequency. FIG. 6 is a viewillustrating a structure of a battery having a multilayer structure. Inthe figure, n represents the number of pairs from the surface of thelaminated body when the conductor layer and the electrolyte layer areregarded as one pair. In this description, it is aimed atthree-dimensional imaging of a battery having a multilayer structure asillustrated in FIG. 6. Magnetic field data at multiple frequencies orimpulse response data of the magnetic field is used.

In the following, a quasi-stationary electromagnetic field equationinside the multilayer structure is derived. It is assumed that theelectrical conductivity is constant inside each layer. Furthermore, itis assumed that the electrical conductivities of the plurality ofelectrolytes are equal to each other, the electrical conductivities ofthe plurality of conductors are equal to each other, and the electricalconductivity of the electrolyte and the electrical conductivity of theconductors are σ_(e) and σ_(c), respectively. The equation about theinside of each layer is represented by Maxwell's equation as follows.∇×E=−∂ _(t) B∇×H=j=σE  (1-1)

Next consideration will be given on the boundary condition between twolayers. Since consideration is given on the case of a quasi-stationaryelectromagnetic field in which the electrical conductivity of aconductor is considered to be finite, it is not necessary to considerthat the current is concentrated on the boundary surface. Accordingly,the following expression is established.1 H _(t) is continuous at boundary surface2 E _(t) is continuous at boundary surface  (1-2)Here, FIG. 7 illustrates the boundary between layers. t represents acomponent in the tangential direction as illustrated in FIG. 7. When thecomponent in the normal direction of the magnetic field is assumed as 0from the assumption of 2D and the expression (1-2) is rewritten usingthe expression (1-1), the expression (1-2) becomes as follows.1 B _(1,t) =B _(2,t)2 ∂_(z) B _(1,t)/σ₁=∂_(z) B _(2,t)/σ₂  (1-3)

Here, the value of magnetic permeability μ is equal for air, aluminum,and copper. From the expression (1-1), if an equation of H only insideeach layer is derived, it becomes

$\begin{matrix}{{\Delta\; H} = {{\sigma\mu}\;\frac{\partial}{\partial t}{H.}}} & \left( {1\text{-}4} \right)\end{matrix}$When this is subjected to Fourier transformation, the followingexpression is obtained.

$\begin{matrix}{{Q\left( {k_{x},k_{y},z,\omega} \right)} = {\int\limits_{- \infty}^{\infty}{\int\limits_{- \infty}^{\infty}{\int\limits_{- \infty}^{\infty}{e^{{{- i}\;\omega\; t} + {{ik}_{x}x} + {{ik}_{y}y}}{H\left( {x,y,z,t} \right)}{dxdydt}}}}}} & \left( {1\text{-}5} \right) \\{{\left( {\frac{d^{2}}{{dz}^{2}} - k_{x}^{2} - k_{y}^{2} - {i\;\omega\;{\sigma\mu}}} \right){Q\left( {k_{x},k_{y},z,\omega} \right)}} = 0} & \left( {1\text{-}6} \right)\end{matrix}$

The general solution of this equation is as follows.Q(k _(x) ,k _(y) ,z,ω)=c ₁(k _(x) ,k _(y),ω)e ^(sz) +c ₂(k _(x) ,k_(y),ω)e ^(−sz)  (1-7)Here, s is as follows.

$\begin{matrix}{\mspace{79mu}{{s^{2} = {k_{x}^{2} + k_{y}^{2} + {i\;\omega\;{\sigma\mu}}}}{s = {\frac{\sqrt{k_{x}^{2} + k_{y}^{2} + \sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\omega^{2}\sigma^{2}\mu^{2}}}}}{\sqrt{2}} + {i\frac{\sqrt{{- k_{x}^{2}} - k_{y}^{2} + \sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\omega^{2}\sigma^{2}\mu^{2}}}}}{\sqrt{2}}}}}}} & \left( {1\text{-}8} \right)\end{matrix}$

With z=0 and z=z in one layer, the following expression is established.Representation with parameters omitted is used.

$\begin{matrix}{{\begin{pmatrix}Q_{0} \\{\overset{.}{Q}}_{0}\end{pmatrix} = {\begin{pmatrix}1 & 1 \\s & {- s}\end{pmatrix}\begin{pmatrix}c_{1} \\c_{2}\end{pmatrix}}}{\begin{pmatrix}Q_{z} \\{\overset{.}{Q}}_{z}\end{pmatrix} = {\begin{pmatrix}e^{sz} & e^{- {sz}} \\{se}^{sz} & {- {se}^{- {sz}}}\end{pmatrix}\begin{pmatrix}c_{1} \\c_{2}\end{pmatrix}}}} & \left( {1\text{-}9} \right)\end{matrix}$

Here, a dot symbol{dot over (Q)}represents differential of Q with respect to z. When (c₁, c₂) is deletedfrom this expression, the following expression is obtained.

$\begin{matrix}{\begin{pmatrix}Q_{z} \\{\overset{.}{Q}}_{z}\end{pmatrix} = {{\begin{pmatrix}e^{sz} & e^{- {sz}} \\{se}^{sz} & {- {se}^{- {sz}}}\end{pmatrix}\begin{pmatrix}{1/2} & {1/\left( {2\; s} \right)} \\{1/2} & {{- 1}/\left( {2s} \right)}\end{pmatrix}\begin{pmatrix}Q_{0} \\{\overset{.}{Q}}_{0}\end{pmatrix}} = {\begin{pmatrix}{\cosh({sz})} & {{{\sinh({sz})}/s}\;} \\{s \cdot {\sinh({sz})}} & {\cosh({sz})}\end{pmatrix}\begin{pmatrix}Q_{0} \\{\overset{.}{Q}}_{0}\end{pmatrix}}}} & \left( {1\text{-}10} \right)\end{matrix}$

This expression is applied to the case where the conductor layer and theelectrolyte layer are paired as illustrated in FIG. 8. FIG. 8 is a viewillustrating definitions of variables in the laminated body. In thisfigure, m indicates the number of pairs from the surface of thelaminated body.

$\begin{matrix}{{\begin{pmatrix}Q_{4} \\{\overset{.}{Q}}_{4}\end{pmatrix} = {\begin{pmatrix}1 & 0 \\0 & \frac{\sigma_{c}}{\sigma_{e}}\end{pmatrix}\begin{pmatrix}Q_{3} \\\overset{.}{Q_{3}}\end{pmatrix}}}{\begin{pmatrix}Q_{3} \\{\overset{.}{Q}}_{3}\end{pmatrix} = {\begin{pmatrix}{\cosh\left( {s_{e}g_{e}} \right)} & {{\sinh\left( {s_{e}g_{e}} \right)}/s_{e}} \\{s_{e} \cdot {\sinh\left( {s_{e}g_{e}} \right)}} & {\cosh\left( {s_{e}g_{e}} \right)}\end{pmatrix}\begin{pmatrix}Q_{2} \\{\overset{.}{Q}}_{2}\end{pmatrix}}}{\begin{pmatrix}Q_{2} \\{\overset{.}{Q}}_{2}\end{pmatrix} = {\begin{pmatrix}1 & 0 \\0 & \frac{\sigma_{e}}{\sigma_{c}}\end{pmatrix}\begin{pmatrix}Q_{1} \\{\overset{.}{Q}}_{1}\end{pmatrix}}}{\begin{pmatrix}Q_{1} \\{\overset{.}{Q}}_{1}\end{pmatrix} = {\begin{pmatrix}{{\cosh\left( {s_{c}g_{c}} \right)}\;} & {{\sinh\left( {s_{c}g_{c}} \right)}/s_{c}} \\{s_{c} \cdot {\sinh\left( {s_{c}g_{c}} \right)}} & {\cosh\left( {s_{c}g_{c}} \right)}\end{pmatrix}\begin{pmatrix}Q_{0} \\{\overset{.}{Q}}_{0}\end{pmatrix}}}} & \left( {1\text{-}11} \right)\end{matrix}$Here, it is assumed that each layer is thin. Specifically, it is assumedthat the following expression is established.

$\begin{matrix}{{{s_{e}g_{e}} \approx {\frac{k_{x}g_{e}}{\sqrt{2}}{\operatorname{<<}1}}},{{s_{c}g_{c}} \approx {\frac{k_{x}g_{c}}{\sqrt{2}}{\operatorname{<<}1}}}} & \left( {1\text{-}12} \right)\end{matrix}$Here, it isk _(x) ≈k _(y).

Leave only the first-order terms of g_(c) and g_(e) in the inside of thematrix.

$\begin{matrix}{{\begin{pmatrix}Q_{4} \\{\overset{.}{Q}}_{4}\end{pmatrix} = {\begin{pmatrix}1 & 0 \\0 & \frac{\sigma_{c}}{\sigma_{e}}\end{pmatrix}\begin{pmatrix}Q_{3} \\{\overset{.}{Q}}_{3}\end{pmatrix}}}{\begin{pmatrix}Q_{3} \\{\overset{.}{Q}}_{3}\end{pmatrix} = {\begin{pmatrix}1 & g_{e} \\{s_{e}^{2}g_{e}} & 1\end{pmatrix}\begin{pmatrix}Q_{2} \\{\overset{.}{Q}}_{2}\end{pmatrix}}}{\begin{pmatrix}Q_{2} \\{\overset{.}{Q}}_{2}\end{pmatrix} = {\begin{pmatrix}1 & 0 \\0 & \frac{\sigma_{e}}{\sigma_{c}}\end{pmatrix}\begin{pmatrix}Q_{1} \\{\overset{.}{Q}}_{1}\end{pmatrix}}}{\begin{pmatrix}Q_{1} \\{\overset{.}{Q}}_{1}\end{pmatrix} = {\begin{pmatrix}1 & g_{c} \\{s_{c}^{2}g_{c}} & 1\end{pmatrix}\begin{pmatrix}Q_{0} \\{\overset{.}{Q}}_{0}\end{pmatrix}}}} & \left( {1\text{-}13} \right)\end{matrix}$If the expression (1-13) is cleaned up, it becomes as follows. Termswithin { } are second-order minute amounts and are ignored.

$\begin{matrix}{\begin{pmatrix}Q_{4} \\{\overset{.}{Q}}_{4}\end{pmatrix} = {\begin{pmatrix}{1 + \left\{ \frac{\sigma_{e}g_{e}g_{c}s_{c}^{2}}{\sigma_{c}} \right\}} & \frac{{\sigma_{c}g_{c}} + {\sigma_{e}g_{e}}}{\sigma_{c}} \\\frac{{\sigma_{c}s_{e}^{2}g_{e}} + {\sigma_{e}s_{c}^{2}g_{c}}}{\sigma_{e}} & {1 + \left\{ \frac{\sigma_{c}g_{c}g_{e}s_{e}^{2}}{\sigma_{e}} \right\}}\end{pmatrix}\begin{pmatrix}Q_{0} \\{\overset{.}{Q}}_{0}\end{pmatrix}}} & \left( {1\text{-}14} \right)\end{matrix}$

The length Δz of one unit is given by the following expression.Δz=g _(c) +g _(e)  (1-15)

Under the preparation described above, a macro (averaged) equationrelating to the horizontal component of the magnetic field is derived.The following expression is established at an extreme limit where Δz issmall. Here,∂/∂zrepresents the differential of the macro.

$\begin{matrix}{{\frac{\partial}{\partial z}\begin{pmatrix}Q_{4} \\{\overset{.}{Q}}_{4}\end{pmatrix}} = {{\lim\limits_{\Delta_{z}\rightarrow 0}{\frac{1}{\Delta\; z}\left\lbrack {\begin{pmatrix}Q_{4} \\{\overset{.}{Q}}_{4}\end{pmatrix} - \begin{pmatrix}Q_{0} \\{\overset{.}{Q}}_{0}\end{pmatrix}} \right\rbrack}} = {{\lim\limits_{{\Delta\; z}\rightarrow 0}{\frac{1}{\Delta\; z}\begin{pmatrix}0 & \frac{{\sigma_{c}g_{c}} + {\sigma_{e}g_{e}}}{\sigma_{c}} \\\frac{{\sigma_{c}s_{e}^{2}g_{e}} + {\sigma_{e}s_{c}^{2}g_{c}}}{\sigma_{e}} & 0\end{pmatrix}\begin{pmatrix}Q_{0} \\{\overset{.}{Q}}_{0}\end{pmatrix}}} = {\begin{pmatrix}0 & \frac{{\sigma_{c}g_{c}} + {\sigma_{e}g_{e}}}{\sigma_{c}\left( {g_{c} + g_{e}} \right)} \\\frac{{\sigma_{c}s_{e}^{2}g_{e}} + {\sigma_{e}s_{c}^{2}g_{c}}}{\sigma_{e}\left( {g_{c} + g_{e}} \right)} & 0\end{pmatrix}\begin{pmatrix}Q_{0} \\{\overset{.}{Q}}_{0}\end{pmatrix}}}}} & \left( {1\text{-}16} \right)\end{matrix}$

It can be written like the following expression.

$\begin{matrix}{{{\frac{\partial}{\partial z}Q_{4}} = {\frac{{\sigma_{c}g_{c}} + {\sigma_{e}g_{e}}}{\sigma_{c}\left( {g_{c} + g_{e}} \right)}{\overset{.}{Q}}_{0}}}{{\frac{\partial}{\partial z}{\overset{.}{Q}}_{4}} = {\frac{{\sigma_{c}s_{e}^{2}g_{e}} + {\sigma_{e}s_{c}^{2}g_{c}}}{\sigma_{e}\left( {g_{c} + g_{e}} \right)}Q_{0}}}} & \left( {1\text{-}17} \right)\end{matrix}$Here, when it isQ ₄ =Q ₀ =φ,{dot over (Q)} ₄ ={dot over (Q)} ₀=ϕthe expression (1-17) becomes as follows.

$\begin{matrix}{{{\frac{\partial}{\partial z}\varphi} = {\frac{{\sigma_{c}g_{c}} + {\sigma_{e}g_{e}}}{\sigma_{c}\left( {g_{c} + g_{e}} \right)}\phi}}{{\frac{\partial}{\partial x}\phi} = \frac{{\sigma_{c}s_{e}^{2}g_{e}} + {\sigma_{e}s_{c}^{2}g_{c}}}{\sigma_{e}\left( {g_{c} + g_{e}} \right)}}} & \left( {1\text{-}18} \right)\end{matrix}$Whenϕis deleted, the following expression is obtained.

$\begin{matrix}{{\frac{\partial^{2}}{\partial z^{2}}\varphi} = {\frac{{\sigma_{c}g_{c}} + {\sigma_{e}g_{e}}}{\sigma_{c}\left( {g_{c} + g_{e}} \right)}\frac{{\sigma_{c}s_{e}^{2}g_{e}} + {\sigma_{e}s_{c}^{2}g_{c}}}{\sigma_{e}\left( {g_{c} + g_{e}} \right)}\varphi}} & \left( {1\text{-}19} \right)\end{matrix}$When the expression (1-8) is substituted into S_(e) and S_(c), thefollowing expression is obtained.

$\begin{matrix}{{\frac{\partial^{2}}{\partial z^{2}}\varphi} = {{\frac{\left( {{\sigma_{c}g_{e}} + {\sigma_{e}g_{c}}} \right)\left( {{\sigma_{c}g_{c}} + {\sigma_{e}g_{e}}} \right)}{\sigma_{c}{\sigma_{e}\left( {g_{c} + g_{e}} \right)}^{2}}\left( {k_{x}^{2} + k_{y}^{2}} \right)\varphi} + {i\;{\omega\mu}\frac{\left( {{\sigma_{c}g_{c}} + {\sigma_{e}g_{e}}} \right)}{\left( {g_{c} + g_{e}} \right)}\varphi}}} & \left( {1\text{-}20} \right)\end{matrix}$

Three-dimensional diffusion equations are obtained by inverse Fouriertransformation.

$\begin{matrix}{{\mu\frac{\partial}{\partial t}\Phi} = {\left\{ {{\frac{\left( {{\sigma_{c}g_{e}} + {\sigma_{e}g_{c}}} \right)}{\sigma_{c}{\sigma_{e}\left( {g_{c} + g_{e}} \right)}}\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}} \right)} + {\frac{g_{c} + g_{e}}{{\sigma_{c}g_{c}} + {\sigma_{e}g_{e}}}\frac{\partial^{2}\;}{\partial z^{2}}}} \right\}\Phi}} & \left( {1\text{-}21} \right)\end{matrix}$Φ is an inverse Fourier transformation ofφ.

<II-2 Boundary Condition Between Air and Battery>

Two sets of boundary conditions are required for each vector element toconnect air and metal electrodes of the battery. For that purpose, theintegral type of Maxwell's equation is used.

$\begin{matrix}{{{\int{\int_{D}{\bigtriangledown \times E\;{dxdy}}}} = {{\oint_{\partial D}{E \cdot {ds}}} = {- {\partial_{t}{\int{\int_{D}{B\;{dxdy}}}}}}}}{{\int{\int_{D}{\bigtriangledown \times H\;{dxdy}}}} = {{\oint_{\partial D}{H \cdot {ds}}} = {\int{\int_{D}{\sigma\; E\;{dxdy}}}}}}} & \left( {2\text{-}1} \right)\end{matrix}$

D is two-dimensional area, ∂D is a counterclockwise boundary

If a rectangular area that is thin in the z-direction as the area D isdefined with the boundary surface interposed therebetween and doubleintegrals are made to approach 0 first, a condition that the tangentialcomponents E_(t) and H_(t) are continuous at the boundary surface isobtained. The electric field is not intended to be included in theboundary condition of the equation represented only by the magneticfield. However, in a case where the electrical conductivity is 0, it is∇×H=j=σE  (2-2)from the second expression of the expression (1-1), and it is impossibleto obtain the condition relating to the differential of H. Accordingly,air is virtually considered as a substance having a small but finiteelectrical conductivity, not a medium with electrical conductivity of 0and its electrical conductivity is assumed to be σ_(a). By doing this,the boundary condition that E_(t) is continuous is represented asfollows.

$\begin{matrix}{{\frac{\bigtriangledown \times H}{\sigma_{a}}{|_{t}}_{air}} = \left. \frac{\bigtriangledown \times H}{\sigma_{c}} \right|_{t_{\underset{electrode}{battery}}}} & \left( {2\text{-}3} \right)\end{matrix}$This expression can be rewritten as follows.

$\begin{matrix}{{{{\sigma_{a}^{- 1}\left( {\frac{\partial H_{z}}{\partial y} - \frac{\partial H_{y}}{\partial z}} \right)}}_{air} = {\sigma_{c}^{- 1}\left( {\frac{\partial H_{z}}{\partial y} - \frac{\partial H_{y}}{\partial z}} \right)}}}_{\overset{battery}{electrode}} & \left( {2\text{-}4} \right) \\{{{{\sigma_{a}^{- 1}\left( {\frac{\partial H_{z}}{\partial x} - \frac{\partial H_{x}}{\partial z}} \right)}}_{air} = {\sigma_{c}^{- 1}\left( {\frac{\partial H_{z}}{\partial x} - \frac{\partial H_{x}}{\partial z}} \right)}}}_{\overset{battery}{electrode}} & \;\end{matrix}$

From the assumption of the 2D magnetic field,

$\begin{matrix}{{{{\sigma_{a}^{- 1}\frac{\partial H_{y}}{\partial z}}}_{air} = {\sigma_{c}^{- 1}\frac{\partial H_{y}}{\partial z}}}}_{\overset{battery}{electrode}} & \left( {2\text{-}5} \right) \\{{{{\sigma_{a}^{- 1}\frac{\partial H_{x}}{\partial z}}}_{air} = {\sigma_{c}^{- 1}\frac{\partial H_{x}}{\partial z}}}}_{\overset{battery}{electrode}} & \;\end{matrix}$is obtained. The magnetic field is subjected to the Fouriertransformation as follows.

$\begin{matrix}{{\overset{\sim}{H}\left( {k_{x},k_{y},z,\omega} \right)} = {\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{e^{{{- i}\;\omega\; t} + {{ik}_{x}x} + {{ik}_{y}y}}{H\left( {x,y,z,t} \right)}{dxdydt}}}}}} & \left( {2\text{-}6} \right)\end{matrix}$

From the expression (2-5), the following expression is obtained.

$\begin{matrix}{{{{\sigma_{a}^{- 1}\frac{\partial{\overset{\sim}{H}}_{y}}{\partial z}}}_{air} = {\sigma_{c}^{- 1}\frac{\partial{\overset{\sim}{H}}_{y}}{\partial z}}}}_{\overset{battery}{electrode}} & \left( {2\text{-}7} \right) \\{{{{\sigma_{a}^{- 1}\frac{\partial{\overset{\sim}{H}}_{x}}{\partial z}}}_{air} = {\sigma_{c}^{- 1}\frac{\partial{\overset{\sim}{H}}_{x}}{\partial z}}}}_{\overset{battery}{electrode}} & \;\end{matrix}$In this way, the boundary conditions were obtained. When these aresummarized once again, it is represented as follows.

$\begin{matrix}{{{{\overset{\sim}{H}}_{x}}_{\overset{battery}{electrode}} = {\overset{\sim}{H}}_{x}}}_{air} & \left( {2\text{-}8} \right) \\{{{{{{{{\overset{\sim}{H}}_{y}}_{\overset{battery}{electrode}} = {\overset{\sim}{H}}_{y}}}_{air}\frac{\partial{\overset{\sim}{H}}_{x}}{\partial z}}}_{\overset{battery}{electrode}} = {\frac{\sigma_{c}}{\sigma_{a}}\frac{\partial{\overset{\sim}{H}}_{x}}{\partial z}}}}_{air} & \; \\{{{\frac{\partial{\overset{\sim}{H}}_{y}}{\partial z}}_{\overset{battery}{electrode}} = {\frac{\sigma_{c}}{\sigma_{a}}\frac{\partial{\overset{\sim}{H}}_{y}}{\partial z}}}}_{air} & \;\end{matrix}$

Since it is assumed that air is a substance having a virtually non-zeroelectrical conductivity, consideration should be given on matters thatthe following expression is established.

$\begin{matrix}{{\Delta\; H} = {\sigma_{a}\mu\frac{\partial}{\partial t}H}} & \left( {2\text{-}9} \right)\end{matrix}$When this is subjected to the Fourier transformation, it becomes asfollows.

$\begin{matrix}{{\left( {\frac{d^{2}}{{dz}^{2}} - k_{x}^{2} - k_{y}^{2} - {i\;\omega\;\sigma_{a}\mu}} \right)\overset{\sim}{H}} = 0} & \left( {2\text{-}10} \right)\end{matrix}$

The general solution of this equation is represented as follows.{tilde over (H)}(k _(x) ,k _(y) ,z,ω)=a(k _(x) ,k _(y),ω)e ^(sz) +b(k_(x) ,k _(y),ω)e ^(−sz)  (2-11)Here, s is as follows.

$\begin{matrix}{\mspace{79mu}{{s^{2} = {k_{x}^{2} + k_{y}^{2} + {i\;\omega\;\sigma\;\mu}}}{s = {\frac{\sqrt{k_{x}^{2} + k_{y}^{2} + \sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\omega^{2}\sigma^{2}\mu^{2}}}}}{\sqrt{2}} + {i\;\frac{\sqrt{{- k_{x}^{2}} - k_{y}^{2} + \sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\omega^{2}\sigma^{2}\mu^{2}}}}}{\sqrt{2}}}}}}} & \left( {2\text{-}12} \right)\end{matrix}$

a and b are obtained by measuring H=(H_(x), H_(y), 0) on two planes inthe air, each of which has a constant z-coordinate. If the expression(2-11) is used, it is possible to calculate the magnetic field at theboundary surface and its differential required for the boundarycondition expression (2-8).

<II-3 Analysis by Inverse Problem Inside Battery>

A target of the inverse problem is written once again. A governingequation of a phenomenon which becomes the target is as follows.

$\begin{matrix}{{\mu\frac{\partial}{\partial t}\Phi} = {\left\{ {{\frac{\left( {{\sigma_{c}g_{e}} + {\sigma_{e}g_{c}}} \right)}{\sigma_{c}{\sigma_{e}\left( {g_{c} + g_{e}} \right)}}\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}} \right)} + {\frac{g_{c} + g_{e}}{{\sigma_{c}g_{c}} + {\sigma_{e}g_{e}}}\frac{\partial^{2}}{\partial ϛ^{2}}}} \right\}\Phi}} & \left( {3\text{-}1} \right)\end{matrix}$The differential such as the expression (1-16) is represented byζinstead of z.

$\begin{matrix}{\frac{\partial}{\partial ϛ} = {\frac{{\sigma_{c}g_{c}} + {\sigma_{e}g_{e}}}{\sigma_{c}\left( {g_{c} + g_{e}} \right)}\frac{\partial}{\partial z}}} & \left( {3\text{-}2} \right)\end{matrix}$φ_(a)(k _(x) ,k _(y),0,ω)=Q _(a)(k _(x) ,k _(y),0,ω)

is representative of the value (H_(x), H_(y)) of the magnetic field inthe air and a known function. From the equation (1-18), the boundarycondition is as follows.

$\begin{matrix}{{{H_{x}}_{\overset{battery}{electrode}} = H_{x}}}_{air} & \left( {3\text{-}3} \right) \\{{{H_{y}}_{\overset{battery}{electrode}} = H_{y}}}_{air} & \; \\{{{{\frac{\partial}{\partial ϛ}H_{x}}}_{\overset{battery}{electrode}} = {\frac{{\sigma_{c}g_{c}} + {\sigma_{e}g_{e}}}{\sigma_{a}\left( {g_{c} + g_{e}} \right)}\frac{\partial}{\partial z}H_{x}}}}_{air} & \; \\{{{{\frac{\partial}{\partial ϛ}H_{y}}}_{\overset{battery}{electrode}} = {\frac{{\sigma_{c}g_{c}} + {\sigma_{e}g_{e}}}{\sigma_{a}\left( {g_{c} + g_{e}} \right)}\frac{\partial}{\partial z}H_{y}}}}_{air} & \; \\{\Phi\left( {x,y,\zeta,t} \right)} & \;\end{matrix}$in the expression (3-1) is the value of H_(x) or H_(y) on the surface ofthe conductor layer.

$\begin{matrix}{{\varphi\left( {k_{x},k_{y},z,\omega} \right)} = {\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{e^{{{- i}\;\omega\; t} + {{ik}_{x}x} + {{ik}_{y}y}}{\Phi\left( {x,y,z,t} \right)}{dxdydt}}}}}} & \left( {3\text{-}4} \right)\end{matrix}$When the expression (3-1) is subjected to the Fourier transformation,

$\begin{matrix}{{\left\{ {{\frac{g_{c} + g_{e}}{{\sigma_{c}g_{c}} + {\sigma_{e}g_{e}}}\frac{d^{2}}{d\; ϛ^{2}}} - {\frac{\left( {{\sigma_{c}g_{e}} + {\sigma_{e}g_{c}}} \right)}{\sigma_{c}{\sigma_{e}\left( {g_{c} + g_{e}} \right)}}\left( {k_{x}^{2} + k_{y}^{2}} \right)} - {i\;\omega\;\mu}} \right\}\varphi} = 0} & \left( {3\text{-}5} \right) \\{{{\frac{g_{c} + g_{e}}{{\sigma_{c}g_{c}} + {\sigma_{e}g_{e}}}s^{2}} - {\frac{\left( {{\sigma_{c}g_{e}} + {\sigma_{e}g_{c}}} \right)}{\sigma_{c}{\sigma_{e}\left( {g_{c} + g_{e}} \right)}}\left( {k_{x}^{2} + k_{y}^{2}} \right)} - {i\;\omega\;\mu}} = 0} & \left( {3\text{-}6} \right)\end{matrix}$are obtained, and when this solution is set as s, the solution of theabove expression can be written as follows.φ(k _(x) ,k _(y) ,z,ω)=a(k _(x) ,k _(y),ω)e ^(sζ) +b(k _(x) ,k _(y),ω)e^(−sζ)   (3-7)Specifically, s is given by the following expression.

$\begin{matrix}{s = {\frac{\sqrt{\begin{matrix}{\frac{\left( {{\sigma_{c}g_{c}} + {\sigma_{e}g_{e}}} \right)^{2}\left( {k_{x}^{2} + k_{y}^{2}} \right)}{\sigma_{c}{\sigma_{e}\left( {g_{c} + g_{e}} \right)}^{2}} +} \\\sqrt{\left( \frac{\left( {{\sigma_{c}g_{c}} + {\sigma_{e}g_{e}}} \right)^{2}\left( {k_{x}^{2} + k_{y}^{2}} \right)}{\sigma_{c}{\sigma_{e}\left( {g_{c} + g_{e}} \right)}^{2}} \right)^{2} + \left( \frac{\omega\;{\mu\left( {{\sigma_{c}g_{c}} + {\sigma_{e}g_{e}}} \right)}}{g_{c} + g_{e}} \right)^{2}}\end{matrix}}}{\sqrt{2}} + {i\frac{\sqrt{\begin{matrix}{{- \frac{\left( {{\sigma_{c}g_{c}} + {\sigma_{e}g_{e}}} \right)^{2}\left( {k_{x}^{2} + k_{y}^{2}} \right)}{\sigma_{c}{\sigma_{e}\left( {g_{c} + g_{e}} \right)}^{2}}} +} \\\sqrt{\left( \frac{\left( {{\sigma_{c}g_{c}} + {\sigma_{e}g_{e}}} \right)^{2}\left( {k_{x}^{2} + k_{y}^{2}} \right)}{\sigma_{c}{\sigma_{e}\left( {g_{c} + g_{e}} \right)}^{2}} \right)^{2} + \left( \frac{\omega\;{\mu\left( {{\sigma_{c}g_{c}} + {\sigma_{e}g_{e}}} \right)}}{g_{c} + g_{e}} \right)^{2}}\end{matrix}}}{\sqrt{2}}}}} & \left( {3\text{-}8} \right)\end{matrix}$

The a and b are determined using the boundary condition formula (3-3).When it is assumed that theζcoordinate of the surface of the battery isζ=0the following expression is obtained.a(k _(x) ,k _(y),ω)+b(k _(x) ,k _(y),ω)=φ(k _(x) ,k _(y),0,ω)sa(k _(x) ,k _(y),ω)−sb(k _(x) ,k _(y),ω)=∂_(ζ)φ(k _(x) ,k_(y),0,ω)  (3-9)The following expression is obtained by solving this.

$\begin{matrix}{{a\left( {k_{x},k_{y},\omega} \right)} = {\frac{1}{2s}\left\{ {{s\;{\varphi\left( {k_{x},k_{y},0,\omega} \right)}} + {\partial_{ϛ}{\varphi\left( {k_{x},k_{y},0,\omega} \right)}}} \right\}}} & \left( {3\text{-}10} \right) \\{{b\left( {k_{x},k_{y},\omega} \right)} = {\frac{1}{2s}\left\{ {{s\;{\varphi\left( {k_{x},k_{y},0,\omega} \right)}} - {\partial_{ϛ}{\varphi\left( {k_{x},k_{y},0,\omega} \right)}}} \right\}}} & \;\end{matrix}$

The expression (3-7) is subjected to the inverse Fourier transformation.

$\begin{matrix}{{\Phi\left( {x,y,Ϛ,t} \right)} = {\frac{1}{\left( {2\pi} \right)^{3}}{\int\limits_{- \infty}^{\infty}{\int\limits_{- \infty}^{\infty}{\int\limits_{- \infty}^{\infty}{e^{{i\;{\omega t}} - {{ik}_{x}x} - {{ik}_{y}y}}{\varphi\left( {k_{x},k_{y},Ϛ,\omega} \right)}{dk}_{x}{dk}_{y}d\;\omega}}}}}} & \left( {3\text{-}11} \right)\end{matrix}$The solution ρ of the inverse problem is given by the followingexpression.

$\begin{matrix}{{\rho\left( {x,y,Ϛ} \right)} = {\lim\limits_{t\rightarrow 0}{\Phi\left( {x,y,Ϛ,t} \right)}}} & \left( {3\text{-}12} \right)\end{matrix}$

<II-4 Consideration>

Consider the expression (1-4).

$\begin{matrix}{{\Delta\; H} = {{\sigma\mu}\;\frac{\partial}{\partial t}H}} & \left( {4\text{-}1} \right)\end{matrix}$When the flux of the magnetic field as in the following expression isdefined, the conductivity λ is 1/σμ.

$\begin{matrix}{\begin{pmatrix}F_{x} \\F_{y}\end{pmatrix} = {{- \frac{1}{\sigma\mu}}{\nabla\begin{pmatrix}H_{x} \\H_{y}\end{pmatrix}}}} & \left( {4\text{-}2} \right)\end{matrix}$In this case, a continuous expression can be written as follows.

$\begin{matrix}{{{\frac{\partial}{\partial t}\begin{pmatrix}H_{x} \\H_{y}\end{pmatrix}} + {\nabla\begin{pmatrix}F_{x} \\F_{y}\end{pmatrix}}} = 0} & \left( {4\text{-}3} \right)\end{matrix}$This is a partial differential equation of the expression (4-1). Fromthese physical considerations, it can be seen that the boundarycondition is a condition that the flux of the magnetic field iscontinuous. Since it can be interpreted asλ=∞in the air,

$\begin{matrix}{{{\nabla_{z}\begin{pmatrix}H_{x} \\H_{y}\end{pmatrix}}} = \infty} & \left( {4\text{-}4} \right)\end{matrix}$should be established at the boundary between the conductor and the air.However, this is a discussion under the condition that the zdifferential of the magnetic field in air is not zero. However, sincethe magnetic field in the air is represented by a harmonic function andthe that regarding z is represented by a simple exponential function,when the z differential becomes zero at the boundary surface with theconductor, the z-component of the magnetic field becomes zero in theentire space. Thus, the above expression (4-4) is correct.

Analyzing the inverse problem using the expression (4-4) is somewhatcumbersome due to singularity. Accordingly, air is considered to be amaterial having a finite minute electrical conductivity, using a penaltymethod. In this case, since the magnetic field is represented by aregular function having no singularity in the entire space which becomesan analysis target, it is easy to handle and it is also evident that themagnetic field approaches the case of ideal air when the virtualelectrical conductivity approaches zero.

<III Basic Theory Assumption of 3D Magnetic Field (B_(y), B_(y), B_(z))>

<III-1 Averaged Basic Equation>

A practical battery has a multilayer structure. Since the existingmagnetic imaging technology is a two-dimensional (x-y plane) imagingtechnology, it was difficult to find out which layer of the multilayerthere is an abnormal portion. Image reconstruction has been performedusing measured data with DC or at a fixed frequency. In thisdescription, it is aimed at three-dimensional imaging of a batteryhaving a multilayer structure as illustrated in FIG. 6. Magnetic fielddata at multiple frequencies or impulse response data of the magneticfield is used.

In the following, a quasi-stationary electromagnetic field equationinside the multilayer structure is derived. It is assumed that theelectrical conductivity is constant inside each layer. Furthermore, itis assumed that the electrical conductivities of the plurality ofelectrolytes are equal to each other, the electrical conductivities ofthe plurality of conductors are equal to each other, and the electricalconductivity of the electrolyte and the electrical conductivity of theconductors are σ_(e) and σ_(e), respectively. The equation about theinside of each layer is represented by Maxwell's equation as follows.∇×E=−∂ _(t) B∇×H=j=σE  (5-1)

Next consideration will be given on the boundary condition between twolayers. Since consideration is given on the case of a quasi-stationaryelectromagnetic field in which the electrical conductivity of aconductor is considered to be finite, it is not necessary to considerthat the current is concentrated on the boundary surface. Accordingly,the following expression is established.1 H _(t) is continuous at boundary surface2 E _(t) is continuous at boundary surface  (5-2)Here, FIG. 9 is a view illustrating a boundary between a layer and alayer. As illustrated in FIG. 9, H_(1,x), H_(1,y), H_(2,x), and H_(2,y)represent the components in the tangential direction of the interface,and H_(1,z) and H_(2,z) indicate the components in the normal directionof the interface, respectively. When it is assumed that the component inthe normal direction of the magnetic field is not 0 from the assumptionof 3D, it is rewritten by using the expression (5-1), and div (H)=0 isfurther considered, the expression (5-2) becomes as follows.1H _(1,x) =H _(2,x)H _(1,y) =H _(2,y)H _(1,z) =H _(2,z)2(∂_(y) H _(1,z) −∂H _(1,y))/σ₁=(∂_(y) H _(2,z)−∂_(z) H _(2,y))/σ₂(∂_(z) H _(1,x)−∂_(x) H _(1,z))/σ₁=(∂_(z) H _(2,x)−∂_(x) H _(2,z))/σ₂∂_(z) H _(1,z)=∂_(z) H _(2,z)  (5-3)

An equation of H only inside each layer derived from the expression(5-1) is

$\begin{matrix}{{\Delta\; H} = {{\sigma\mu}\;\frac{\partial}{\partial t}H}} & \left( {5\text{-}4} \right)\end{matrix}$and when this is subjected to Fourier transformation, the followingexpression is obtained.

$\begin{matrix}{{Q\left( {k_{x},k_{y},z,\omega} \right)} = {\int\limits_{- \infty}^{\infty}{\int\limits_{- \infty}^{\infty}{\int\limits_{- \infty}^{\infty}{e^{{{- i}\;\omega\; t} + {{ik}_{x}x} + {{ik}_{y}y}}{H\left( {x,y,z,t} \right)}{dxdydt}}}}}} & \left( {5\text{-}5} \right) \\{{\left( {\frac{d^{2}}{{dz}^{2}} - k_{x}^{2} - k_{y}^{2} - {i\;\omega\;{\sigma\mu}}} \right){Q\left( {k_{x},k_{y},z,\omega} \right)}} = 0} & \left( {5\text{-}6} \right)\end{matrix}$The general solution of this equation is as follows.Q(k _(x) ,k _(y) ,z,ω)=c(k _(x) ,k _(y),ω)e ^(sz) +c ₂(k _(x) ,k_(y),ω)e ^(−sz)   (5-7)c₁ and c₂ are vectors. Here, s is as follows.

$\begin{matrix}{\mspace{79mu}{{s^{2} = {k_{x}^{2} + k_{y}^{2} + {i\;\omega\;\sigma\;\mu}}}{s = {\frac{\sqrt{k_{x}^{2} + k_{y}^{2} + \sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\omega^{2}\sigma^{2}\mu^{2}}}}}{\sqrt{2}} + {i\frac{\sqrt{{- k_{x}^{2}} - k_{y}^{2} + \sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\omega^{2}\sigma^{2}\mu^{2}}}}}{\sqrt{2}}}}}}} & \left( {5\text{-}8} \right)\end{matrix}$

The following vector is defined. Here, the dot symbol{dot over (Q)}represents the differential of Q with respect to z.

$\begin{pmatrix}Q \\\overset{.}{Q}\end{pmatrix} = \begin{pmatrix}Q_{x} \\Q_{y} \\Q_{z} \\{\overset{.}{Q}}_{x} \\{\overset{.}{Q}}_{y} \\{\overset{.}{Q}}_{z}\end{pmatrix}$

With z=0 and z=z in one layer, the following expression is established.Representation with parameters omitted is used.

$\begin{matrix}{{\begin{pmatrix}{Q(0)} \\{\overset{.}{Q}(0)}\end{pmatrix} = {\begin{pmatrix}1 & 1 \\s & {- s}\end{pmatrix}\begin{pmatrix}c_{1} \\c_{2}\end{pmatrix}}}{\begin{pmatrix}{Q(z)} \\{\overset{.}{Q}(z)}\end{pmatrix} = {\begin{pmatrix}e^{sz} & e^{- {sz}} \\{se}^{sz} & {- {se}^{sz}}\end{pmatrix}\begin{pmatrix}c_{1} \\c_{2}\end{pmatrix}}}} & \left( {5\text{-}9} \right)\end{matrix}$When (c₁, c₂) is deleted from this expression, the following expressionis obtained.

$\begin{matrix}\begin{matrix}{\begin{pmatrix}{Q(z)} \\{\overset{.}{Q}(z)}\end{pmatrix} = {\begin{pmatrix}e^{sz} & e^{- {sz}} \\{se}^{sz} & {- {se}^{- {sz}}}\end{pmatrix}\begin{pmatrix}{1/2} & {1/\left( {2s} \right)} \\{1/2} & {{- 1}/\left( {2s} \right)}\end{pmatrix}\begin{pmatrix}{Q(0)} \\{\overset{.}{Q}(0)}\end{pmatrix}}} \\{= {\begin{pmatrix}{\cosh({sz})} & {{\sinh({sz})}/s} \\{s \cdot {\sinh({sz})}} & {\cosh({sz})}\end{pmatrix}\begin{pmatrix}{Q(0)} \\{\overset{.}{Q}(0)}\end{pmatrix}}}\end{matrix} & \left( {5\text{-}10} \right)\end{matrix}$

This expression is applied to the case where the conductor layer and theelectrolyte layer are paired as illustrated in FIG. 8. Then, theexpression to connect the magnetic field at each layer can be written asfollows.

$\begin{matrix}{{\begin{pmatrix}Q_{4x} \\Q_{4y} \\Q_{4z} \\{\overset{.}{Q}}_{4x} \\{\overset{.}{Q}}_{4y} \\{\overset{.}{Q}}_{4z}\end{pmatrix} = {\begin{pmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & {{ik}_{x}\left( {{\sigma_{c}/\sigma_{e}} - 1} \right)} & {\sigma_{c}/\sigma_{e}} & 0 & 0 \\0 & 0 & {{ik}_{y}\left( {{\sigma_{c}/\sigma_{e}} - 1} \right)} & 0 & {\sigma_{c}/\sigma_{e}} & 0 \\0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}\begin{pmatrix}Q_{3x} \\Q_{3y} \\Q_{3z} \\{\overset{.}{Q}}_{3x} \\{\overset{.}{Q}}_{3y} \\{\overset{.}{Q}}_{3z}\end{pmatrix}}}\mspace{79mu}{\begin{pmatrix}Q_{3} \\{\overset{.}{Q}}_{3}\end{pmatrix} = {\begin{pmatrix}{\cosh\left( {s_{e}g_{e}} \right)} & {{\sinh\left( {s_{e}g_{e}} \right)}/s_{e}} \\{s_{e} \cdot {\sinh\left( {s_{e}g_{e}} \right)}} & {\cosh\left( {s_{e}g_{e}} \right)}\end{pmatrix}\begin{pmatrix}Q_{2} \\{\overset{.}{Q}}_{2}\end{pmatrix}}}{\begin{pmatrix}Q_{2x} \\Q_{2y} \\Q_{2z} \\{\overset{.}{Q}}_{2x} \\{\overset{.}{Q}}_{2y} \\{\overset{.}{Q}}_{2z}\end{pmatrix} = {\begin{pmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & {{ik}_{x}\left( {{\sigma_{e}/\sigma_{c}} - 1} \right)} & {\sigma_{e}/\sigma_{c}} & 0 & 0 \\0 & 0 & {{ik}_{y}\left( {{\sigma_{e}/\sigma_{c}} - 1} \right)} & 0 & {\sigma_{e}/\sigma_{c}} & 0 \\0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}\begin{pmatrix}Q_{1x} \\Q_{1y} \\Q_{1z} \\{\overset{.}{Q}}_{1x} \\{\overset{.}{Q}}_{1y} \\{\overset{.}{Q}}_{1z}\end{pmatrix}}}\mspace{79mu}{\begin{pmatrix}Q_{1} \\{\overset{.}{Q}}_{1}\end{pmatrix} = {\begin{pmatrix}{\cosh\left( {s_{c}g_{c}} \right)} & {{\sinh\left( {s_{c}g_{c}} \right)}/s_{c}} \\{s_{e} \cdot {\sinh\left( {s_{c}g_{c}} \right)}} & {\cosh\left( {s_{c}g_{c}} \right)}\end{pmatrix}\begin{pmatrix}Q_{0} \\{\overset{.}{Q}}_{0}\end{pmatrix}}}} & \left( {5\text{-}11} \right)\end{matrix}$

Here, the following expression is used.s _(e) ² =k _(x) ² +k _(y) ² +iωσ _(e)μs _(c) ² =k _(x) ² +k _(y) ² +iωσ _(c)μ

Here, it is assumed that each layer is thin. Specifically, it is assumedthat the following expression is established. Here, it isk _(x) ≈k _(y)

$\begin{matrix}{{{s_{e}g_{e}} \approx {\frac{k_{x}g_{e}}{\sqrt{2}}{\operatorname{<<}1}}},{{s_{c}g_{c}} \approx {\frac{k_{x}g_{c}}{\sqrt{2}}{\operatorname{<<}1}}}} & \left( {5\text{-}12} \right)\end{matrix}$

Leave only the first-order terms of g_(c) and g_(e) in the inside of thematrix.

$\begin{matrix}{{\begin{pmatrix}Q_{4x} \\Q_{4y} \\Q_{4z} \\{\overset{.}{Q}}_{4x} \\{\overset{.}{Q}}_{4y} \\{\overset{.}{Q}}_{4z}\end{pmatrix} = {\begin{pmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & {{ik}_{x}\left( {{\sigma_{c}/\sigma_{e}} - 1} \right)} & {\sigma_{c}/\sigma_{e}} & 0 & 0 \\0 & 0 & {{ik}_{y}\left( {{\sigma_{c}/\sigma_{e}} - 1} \right)} & 0 & {\sigma_{c}/\sigma_{e}} & 0 \\0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}\begin{pmatrix}Q_{3x} \\Q_{3y} \\Q_{3z} \\{\overset{.}{Q}}_{3x} \\{\overset{.}{Q}}_{3y} \\{\overset{.}{Q}}_{3z}\end{pmatrix}}}\mspace{79mu}{\begin{pmatrix}Q_{3x} \\Q_{3y} \\Q_{3z} \\{\overset{.}{Q}}_{3x} \\{\overset{.}{Q}}_{3y} \\{\overset{.}{Q}}_{3z}\end{pmatrix} = {\begin{pmatrix}1 & 0 & 0 & g_{e} & 0 & 0 \\0 & 1 & 0 & 0 & g_{e} & 0 \\0 & 0 & 1 & 0 & 0 & g_{e} \\{s_{e}^{2}g_{e}} & 0 & 0 & 1 & 0 & 0 \\0 & {s_{e}^{2}g_{e}} & 0 & 0 & 1 & 0 \\0 & 0 & {s_{e}^{2}g_{e}} & 0 & 0 & 1\end{pmatrix}\begin{pmatrix}Q_{2x} \\Q_{2y} \\Q_{2z} \\{\overset{.}{Q}}_{2x} \\{\overset{.}{Q}}_{2y} \\{\overset{.}{Q}}_{2z}\end{pmatrix}}}{\begin{pmatrix}Q_{2x} \\Q_{2y} \\Q_{2z} \\{\overset{.}{Q}}_{2x} \\{\overset{.}{Q}}_{2y} \\{\overset{.}{Q}}_{2z}\end{pmatrix} = {\begin{pmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & {{ik}_{x}\left( {{\sigma_{e}/\sigma_{c}} - 1} \right)} & {\sigma_{e}/\sigma_{c}} & 0 & 0 \\0 & 0 & {{ik}_{y}\left( {{\sigma_{e}/\sigma_{c}} - 1} \right)} & 0 & {\sigma_{e}/\sigma_{c}} & 0 \\0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}\begin{pmatrix}Q_{1x} \\Q_{1y} \\Q_{1z} \\{\overset{.}{Q}}_{1x} \\{\overset{.}{Q}}_{1y} \\{\overset{.}{Q}}_{1z}\end{pmatrix}}}\mspace{79mu}{\begin{pmatrix}Q_{1x} \\Q_{1y} \\Q_{1z} \\{\overset{.}{Q}}_{1x} \\{\overset{.}{Q}}_{1y} \\{\overset{.}{Q}}_{1z}\end{pmatrix} = {\begin{pmatrix}1 & 0 & 0 & g_{c} & 0 & 0 \\0 & 1 & 0 & 0 & g_{c} & 0 \\0 & 0 & 1 & 0 & 0 & g_{c} \\{s_{c}^{2}g_{c}} & 0 & 0 & 1 & 0 & 0 \\0 & {s_{c}^{2}g_{c}} & 0 & 0 & 1 & 0 \\0 & 0 & {s_{c}^{2}g_{c}} & 0 & 0 & 1\end{pmatrix}\begin{pmatrix}Q_{0x} \\Q_{0y} \\Q_{0z} \\{\overset{.}{Q}}_{0x} \\{\overset{.}{Q}}_{0y} \\{\overset{.}{Q}}_{0z}\end{pmatrix}}}} & \left( {5\text{-}13} \right)\end{matrix}$The following notations are used.

$\begin{matrix}{\mspace{79mu}{{\begin{pmatrix}Q_{4} \\{\overset{.}{Q}}_{4}\end{pmatrix} = {A_{4,2}\begin{pmatrix}Q_{2} \\{\overset{.}{Q}}_{2}\end{pmatrix}}}\mspace{79mu}{\begin{pmatrix}Q_{2} \\{\overset{.}{Q}}_{2}\end{pmatrix} = {A_{2,1}\begin{pmatrix}Q_{0} \\{\overset{.}{Q}}_{0}\end{pmatrix}}}}} & \left( {5\text{-}14} \right) \\{A_{4,2} = {{\begin{pmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & {{ik}_{x}\left( {{\sigma_{c}/\sigma_{e}} - 1} \right)} & {\sigma_{c}/\sigma_{e}} & 0 & 0 \\0 & 0 & {{ik}_{y}\left( {{\sigma_{c}/\sigma_{e}} - 1} \right)} & 0 & {\sigma_{c}/\sigma_{e}} & 0 \\0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}\begin{pmatrix}1 & 0 & 0 & g_{e} & 0 & 0 \\0 & 1 & 0 & 0 & g_{e} & 0 \\0 & 0 & 1 & 0 & 0 & g_{ee} \\{s_{e}^{2}g_{e}} & 0 & 0 & 1 & 0 & 0 \\0 & {s_{e}^{2}g_{e}} & 0 & 0 & 1 & 0 \\0 & 0 & {s_{e}^{2}g_{e}} & 0 & 0 & 1\end{pmatrix}} = \begin{pmatrix}1 & 0 & 0 & g_{e} & 0 & 0 \\0 & 1 & 0 & 0 & g_{e} & 0 \\0 & 0 & 1 & 0 & 0 & g_{e} \\{s_{e}^{2}g_{e}{\sigma_{c}/\sigma_{e}}} & 0 & {{ik}_{x}\left( {{\sigma_{c}/\sigma_{e}} - 1} \right)} & {\sigma_{c}/\sigma_{e}} & 0 & {{ik}_{x}{g_{e}\left( {{\sigma_{c}/\sigma_{e}} - 1} \right)}} \\0 & {s_{e}^{2}g_{e}{\sigma_{c}/\sigma_{e}}} & {{ik}_{y}\left( {{\sigma_{c}/\sigma_{e}} - 1} \right)} & 0 & {\sigma_{c}/\sigma_{e}} & {{ik}_{y}{g_{e}\left( {{\sigma_{c}/\sigma_{e}} - 1} \right)}} \\0 & 0 & {s_{e}^{2}g_{e}} & 0 & 0 & 1\end{pmatrix}}} & \left( {5\text{-}15} \right) \\{A_{2,0} = {{\begin{pmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & {{ik}_{x}\left( {{\sigma_{e}/\sigma_{c}} - 1} \right)} & {\sigma_{e}/\sigma_{c}} & 0 & 0 \\0 & 0 & {{ik}_{y}\left( {{\sigma_{e}/\sigma_{c}} - 1} \right)} & 0 & {\sigma_{e}/\sigma_{c}} & 0 \\0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}\begin{pmatrix}1 & 0 & 0 & g_{c} & 0 & 0 \\0 & 1 & 0 & 0 & g_{c} & 0 \\0 & 0 & 1 & 0 & 0 & g_{c} \\{s_{c}^{2}g_{c}} & 0 & 0 & 1 & 0 & 0 \\0 & {s_{c}^{2}g_{c}} & 0 & 0 & 1 & 0 \\0 & 0 & {s_{c}^{2}g_{c}} & 0 & 0 & 1\end{pmatrix}} = \begin{pmatrix}1 & 0 & 0 & g_{c} & 0 & 0 \\0 & 1 & 0 & 0 & g_{c} & 0 \\0 & 0 & 1 & 0 & 0 & g_{c} \\{s_{c}^{2}g_{c}{\sigma_{e}/\sigma_{c}}} & 0 & {{ik}_{x}\left( {{\sigma_{e}/\sigma_{c}} - 1} \right)} & {\sigma_{e}/\sigma_{c}} & 0 & {{ik}_{x}{g_{c}\left( {{\sigma_{e}/\sigma_{c}} - 1} \right)}} \\0 & {s_{c}^{2}g_{c}{\sigma_{e}/\sigma_{c}}} & {{ik}_{y}\left( {{\sigma_{e}/\sigma_{c}} - 1} \right)} & 0 & {\sigma_{e}/\sigma_{c}} & {{ik}_{y}{g_{c}\left( {{\sigma_{e}/\sigma_{c}} - 1} \right)}} \\0 & 0 & {s_{c}^{2}g_{c}} & 0 & 0 & 1\end{pmatrix}}} & \left( {5\text{-}16} \right)\end{matrix}$The following matrix notation is used.A _(4,0) =A _(4,2) ·A _(2,1)  (5-17)In this case, the following expression is established.

$\begin{matrix}{\mspace{79mu}{\begin{pmatrix}Q_{4} \\{\overset{.}{Q}}_{4}\end{pmatrix} = {A_{4,0}\begin{pmatrix}Q_{0} \\{\overset{.}{Q}}_{0}\end{pmatrix}}}} & \left( {5\text{-}18} \right) \\{A_{4,0} = {{\begin{pmatrix}1 & 0 & 0 & g_{e} & 0 & 0 \\0 & 1 & 0 & 0 & g_{e} & 0 \\0 & 0 & 1 & 0 & 0 & g_{e} \\{s_{e}^{2}g_{e}{\sigma_{c}/\sigma_{e}}} & 0 & {{ik}_{x}\left( {{\sigma_{c}/\sigma_{e}} - 1} \right)} & {\sigma_{c}/\sigma_{e}} & 0 & {{ik}_{x}{g_{e}\left( {{\sigma_{c}/\sigma_{e}} - 1} \right)}} \\0 & {s_{e}^{2}g_{e}{\sigma_{c}/\sigma_{e}}} & {{ik}_{y}\left( {{\sigma_{c}/\sigma_{e}} - 1} \right)} & 0 & {\sigma_{c}/\sigma_{e}} & {{ik}_{y}{g_{e}\left( {{\sigma_{c}/\sigma_{e}} - 1} \right)}} \\0 & 0 & {s_{e}^{2}g_{e}} & 0 & 0 & 1\end{pmatrix} \cdot \begin{pmatrix}1 & 0 & 0 & g_{c} & 0 & 0 \\0 & 1 & 0 & 0 & g_{c} & 0 \\0 & 0 & 1 & 0 & 0 & g_{c} \\{s_{c}^{2}g_{c}{\sigma_{e}/\sigma_{c}}} & 0 & {{ik}_{x}\left( {{\sigma_{e}/\sigma_{c}} - 1} \right)} & {\sigma_{e}/\sigma_{c}} & 0 & {{ik}_{x}{g_{c}\left( {{\sigma_{e}/\sigma_{c}} - 1} \right)}} \\0 & {s_{c}^{2}g_{c}{\sigma_{e}/\sigma_{c}}} & {{ik}_{y}\left( {{\sigma_{e}/\sigma_{c}} - 1} \right)} & 0 & {\sigma_{e}/\sigma_{c}} & {{ik}_{y}{g_{c}\left( {{\sigma_{e}/\sigma_{c}} - 1} \right)}} \\0 & 0 & {s_{c}^{2}g_{c}} & 0 & 0 & 1\end{pmatrix}} = \begin{pmatrix}{1 + {s_{c}^{2}g_{c}g_{e}{\sigma_{e}/\sigma_{c}}}} & 0 & {{ik}_{x}{g_{e}\left( {{\sigma_{e}/\sigma_{c}} - 1} \right)}} & {g_{c} + {g_{e}{\sigma_{e}/\sigma_{c}}}} & 0 & {{ik}_{x}g_{c}{g_{e}\left( {{\sigma_{e}/\sigma_{c}} - 1} \right)}} \\0 & {1 + {s_{c}^{2}g_{c}g_{e}{\sigma_{e}/\sigma_{c}}}} & {{ik}_{y}{g_{e}\left( {{\sigma_{e}/\sigma_{c}} - 1} \right)}} & 0 & {g_{c} + {g_{e}{\sigma_{e}/\sigma_{c}}}} & {{ik}_{y}g_{c}{g_{e}\left( {{\sigma_{e}/\sigma_{c}} - 1} \right)}} \\0 & 0 & {1 + {s_{c}^{2}g_{c}g_{e}}} & 0 & 0 & {g_{c} + g_{e}} \\{{s_{e}^{2}g_{e}{\sigma_{c}/\sigma_{e}}} + {s_{c}^{2}g_{c}}} & 0 & {{ik}_{x}g_{e}g_{c}{s_{c}^{2}\left( {{\sigma_{c}/\sigma_{e}} - 1} \right)}} & {{s_{e}^{2}g_{c}g_{e}{\sigma_{c}/\sigma_{e}}} + 1} & 0 & {{ik}_{x}{g_{e}\left( {{\sigma_{c}/\sigma_{e}} - 1} \right)}} \\0 & {{s_{e}^{2}g_{e}{\sigma_{c}/\sigma_{e}}} + {s_{c}^{2}g_{c}}} & {{ik}_{y}g_{e}g_{c}{s_{c}^{2}\left( {{\sigma_{c}/\sigma_{e}} - 1} \right)}} & 0 & {{s_{e}^{2}g_{e}g_{c}{\sigma_{c}/\sigma_{e}}} + 1} & {{ik}_{y}{g_{e}\left( {{\sigma_{c}/\sigma_{e}} - 1} \right)}} \\0 & 0 & {{s_{e}^{2}g_{e}} + {s_{c}^{2}g_{c}}} & 0 & 0 & {{s_{e}^{2}g_{e}g_{c}} + 1}\end{pmatrix}}} & \left( {5\text{-}19} \right)\end{matrix}$

When ignoring terms of second or higher order on g_(e), g_(c), thefollowing expression is obtained.

$\begin{matrix}{A_{4,0} = \begin{pmatrix}1 & 0 & {{ik}_{x}{g_{e}\left( {{\sigma_{e}/\sigma_{c}} - 1} \right)}} & {g_{c} + {g_{e}{\sigma_{e}/\sigma_{c}}}} & 0 & 0 \\0 & 1 & {{ik}_{y}{g_{e}\left( {{\sigma_{e}/\sigma_{c}} - 1} \right)}} & 0 & {g_{c} + {g_{e}{\sigma_{e}/\sigma_{c}}}} & 0 \\0 & 0 & 1 & 0 & 0 & {g_{c} + g_{e}} \\{{s_{e}^{2}g_{e}{\sigma_{c}/\sigma_{e}}} + {s_{c}^{2}g_{c}}} & 0 & 0 & 1 & 0 & {{ik}_{x}{g_{e}\left( {{\sigma_{c}/\sigma_{e}} - 1} \right)}} \\0 & {{s_{e}^{2}g_{e}{\sigma_{c}/\sigma_{e}}} + {s_{c}^{2}g_{c}}} & 0 & 0 & 1 & {{ik}_{y}{g_{e}\left( {{\sigma_{c}/\sigma_{e}} - 1} \right)}} \\0 & 0 & {{s_{e}^{2}g_{e}} + {s_{c}^{2}g_{c}}} & 0 & 0 & 1\end{pmatrix}} & \left( {5\text{-}20} \right)\end{matrix}$

Under the preparation described above, a macro (averaged) equationrelating to the horizontal component of the magnetic field will bederived. The length Δz of one unit is given by the following expression.Δz=g _(c) +g _(e)  (5-21)

The following expression (5-22) is established at an extreme limit whereΔz is small.∂/∂zrepresents the differential of a macro.

$\begin{matrix}{{\frac{\partial}{\partial z}\begin{pmatrix}Q_{4} \\{\overset{.}{Q}}_{4}\end{pmatrix}} = {\lim\limits_{{\Delta\; z}\rightarrow 0}{\frac{1}{\Delta\; z}\left\lbrack {\begin{pmatrix}Q_{4} \\{\overset{.}{Q}}_{4}\end{pmatrix} - \begin{pmatrix}Q_{0} \\{\overset{.}{Q}}_{0}\end{pmatrix}} \right\rbrack}}} & \left( {5\text{-}22} \right)\end{matrix}$Here, when it isQ ₄ =Q ₀ =φ,{dot over (Q)} ₄ ={dot over (Q)} ₀=ϕ,the following expression is established for each component.

$\begin{matrix}{\mspace{79mu}{{{\left( {g_{c} + g_{e}} \right)\frac{\partial}{\partial z}\varphi_{1}} = {{{ik}_{x}{g_{e}\left( {{\sigma_{e}/\sigma_{c}} - 1} \right)}\varphi_{3}} + {\left( {g_{c} + {g_{e}{\sigma_{e}/\sigma_{c}}}} \right)\phi_{1}}}}\mspace{79mu}{{\left( {g_{c} + g_{e}} \right)\frac{\partial}{\partial z}\varphi_{2}} = {{{ik}_{y}{g_{e}\left( {{\sigma_{e}/\sigma_{c}} - 1} \right)}\varphi_{3}} + {\left( {g_{c} + {g_{e}{\sigma_{e}/\sigma_{c}}}} \right)\phi_{2}}}}\mspace{79mu}{{\left( {g_{c} + g_{e}} \right)\frac{\partial}{\partial z}\varphi_{3}} = {{\left( {g_{c} + g_{e}} \right){\phi_{3}\left( {g_{c} + g_{e}} \right)}\frac{\partial}{\partial z}\phi_{1}} = {{{\left( {{s_{e}^{2}g_{e}{\sigma_{c}/\sigma_{e}}} + {s_{c}^{2}g_{c}}} \right)\varphi_{1}} + {{ik}_{x}{g_{e}\left( {{\sigma_{c}/\sigma_{e}} - 1} \right)}{\phi_{3}\left( {g_{c} + g_{e}} \right)}\frac{\partial}{\partial z}\phi_{2}}} = {{\left( {{s_{e}^{2}g_{e}{\sigma_{c}/\sigma_{e}}} + {s_{c}^{2}g_{c}}} \right)\varphi_{2}} + {{ik}_{y}{g_{e}\left( {{\sigma_{c}/\sigma_{e}} - 1} \right)}\phi_{3}}}}}}\mspace{79mu}{{\left( {g_{c} + g_{e}} \right)\frac{\partial}{\partial z}\phi_{3}} = {\left( {{s_{e}^{2}g_{e}} + {s_{c}^{2}g_{c}}} \right)\varphi_{3}}}}} & \left( {5\text{-}23} \right)\end{matrix}$

When the first, second, and third expressions are differentiated and thefourth, fifth, and sixth expressions are substituted into theexpressions, the following expression is obtained.

$\begin{matrix}{{\left( {g_{c} + g_{e}} \right)^{2}\frac{\partial^{2}}{\partial z^{2}}\varphi_{1}} = {{{\left\lbrack {\left( {g_{c} + {g_{e}{\sigma_{e}/\sigma_{c}}}} \right)\left\{ {{g_{e}{{\sigma_{c}\left( {k_{x}^{2} + k_{y}^{2} + {i\;\omega\;\sigma_{e}\mu}} \right)}/\sigma_{e}}} + {g_{c}\left( {k_{x}^{2} + k_{y}^{2} + {i\;\omega\;\sigma_{c}\mu}} \right)}} \right\}} \right\rbrack\varphi_{1}} + {\left\{ {{{ik}_{x}{g_{e}\left( {g_{c} + g_{e}} \right)}\left( {{\sigma_{e}/\sigma_{c}} - 1} \right)} + {\left( {g_{c} + {g_{e}{\sigma_{e}/\sigma_{c}}}} \right){ik}_{x}{g_{e}\left( {{\sigma_{c}/\sigma_{e}} - 1} \right)}}} \right\}\frac{\partial}{\partial z}{\varphi_{3}\left( {g_{c} + g_{e}} \right)}^{2}\frac{\partial^{2}}{\partial z^{2}}\varphi_{2}}} = {{{\left\lbrack {\left( {g_{c} + {g_{e}{\sigma_{e}/\sigma_{c}}}} \right)\left\{ {{g_{e}{{\sigma_{c}\left( {k_{x}^{2} + k_{y}^{2} + {i\;\omega\;\sigma_{e}\mu}} \right)}/\sigma_{e}}} + {g_{e}\left( {k_{x}^{2} + k_{y}^{2} + {i\;\omega\;\sigma_{c}\mu}} \right)}} \right\}} \right\rbrack\varphi_{2}} + {\left\{ {{{ik}_{y}{g_{e}\left( {g_{c} + g_{e}} \right)}\left( {{\sigma_{e}/\sigma_{c}} - 1} \right)} + {\left( {g_{c} + {g_{e}{\sigma_{e}/\sigma_{c}}}} \right){ik}_{y}{g_{e}\left( {{\sigma_{c}/\sigma_{e}} - 1} \right)}}} \right\}\frac{\partial}{\partial z}{\varphi_{3}\left( {g_{c} + g_{e}} \right)}\frac{\partial^{2}}{\partial z^{2}}\varphi_{3}}} = {\left\{ {{g_{e}\left( {k_{x}^{2} + k_{y}^{2} - {i\;\omega\;\sigma_{e}\mu}} \right)} + {g_{c}\left( {k_{x}^{2} + k_{y}^{2} + {i\;\omega\;\sigma_{c}\mu}} \right)}} \right\}\varphi_{3}}}}} & \left( {5\text{-}24} \right)\end{matrix}$When this is cleaned up,

$\begin{matrix}{{\left( {g_{c} + g_{e}} \right)^{2}\frac{\partial^{2}}{\partial z^{2}}\varphi_{1}} = {{{\left\{ {{\frac{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)\left( {{g_{e}\sigma_{c}} + {g_{c}\sigma_{e}}} \right)}{\sigma_{c}\sigma_{e}}\left( {k_{x}^{2} + k_{y}^{2}} \right)} + {\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)\left( {g_{e} + g_{c}} \right)i\;{\omega\mu}}} \right\}\varphi_{1}} + {\left\{ {{{ik}_{x}{g_{c}\left( {g_{c} + g_{e}} \right)}\left( {{\sigma_{e}/\sigma_{c}} - 1} \right)} + {\left( {g_{c} + {g_{e}{\sigma_{e}/\sigma_{c}}}} \right){ik}_{x}{g_{e}\left( {{\sigma_{c}/\sigma_{e}} - 1} \right)}}} \right\}\frac{\partial}{\partial z}{\varphi_{3}\left( {g_{c} + g_{e}} \right)}^{2}\frac{\partial^{2}}{\partial z^{2}}\varphi_{2}}} = {{{\left\{ {{\frac{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)\left( {{g_{e}\sigma_{c}} + {g_{c}\sigma_{e}}} \right)}{\sigma_{c}\sigma_{e}}\left( {k_{x}^{2} + k_{y}^{2}} \right)} + {\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)\left( {g_{e} + g_{c}} \right)i\;{\omega\mu}}} \right\}\varphi_{2}} + {\left\{ {{{ik}_{y}{g_{e}\left( {g_{c} + g_{e}} \right)}\left( {{\sigma_{e}/\sigma_{c}} - 1} \right)} + {\left( {g_{c} + {g_{e}{\sigma_{e}/\sigma_{c}}}} \right){ik}_{y}{g_{e}\left( {{\sigma_{c}/\sigma_{e}} - 1} \right)}}} \right\}\frac{\partial}{\partial z}{\varphi_{3}\left( {g_{c} + g_{e}} \right)}\frac{\partial^{2}}{\partial z^{2}}\varphi_{3}}} = {\left\{ {{\left( {g_{e} + g_{c}} \right)\left( {k_{x}^{2} + k_{y}^{2}} \right)} + {\left( {{g_{e}\sigma_{e}} + {g_{c}\sigma_{c}}} \right)i\;{\omega\mu}}} \right\}\varphi_{3}}}}} & \left( {5\text{-}25} \right)\end{matrix}$is obtained and three-dimensional diffusion equations are obtained byinverse Fourier transformation.

$\begin{matrix}{{{\mu\frac{\partial}{\partial t}H_{x}} = {{\left\{ {{\frac{\left( {{g_{e}\sigma_{c}} + {g_{c}\sigma_{e}}} \right)}{\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}} \right)} + {\frac{\left( {g_{c} + g_{e}} \right)}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)}\frac{\partial^{2}}{\partial z^{2}}}} \right\} H_{x}} + {\frac{g_{e}{g_{c}\left( {\sigma_{c} - \sigma_{e}} \right)}^{2}}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\frac{\partial^{2}}{{\partial x}{\partial z}}H_{z}}}}{{\mu\frac{\partial}{\partial t}H_{y}} = {{\left\{ {{\frac{\left( {{g_{e}\sigma_{c}} + {g_{c}\sigma_{e}}} \right)}{\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}} \right)} + {\frac{\left( {g_{c} + g_{e}} \right)}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)}\frac{\partial^{2}}{\partial z^{2}}}} \right\} H_{y}} + {\frac{g_{e}{g_{c}\left( {\sigma_{c} - \sigma_{e}} \right)}^{2}}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\frac{\partial^{2}}{{\partial y}{\partial z}}H_{z}}}}\mspace{79mu}{{\mu\frac{\partial}{\partial t}H_{z}} = {\frac{\left( {g_{e} + g_{c}} \right)}{\left( {{g_{e}\sigma_{e}} + {g_{c}\sigma_{c}}} \right)}\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}}} \right)H_{z}}}} & \left( {5\text{-}26} \right)\end{matrix}$The expression (5-26) corresponds to the above expression (11). If it isset that Hz=0 in the first and second of this equation, this equation isthe same as the expression (1-21).

<III-2 Boundary Condition Between Air and Battery>

Two sets of boundary conditions are required for each vector element toconnect air and metal electrodes of the battery. For that purpose, theintegral type of Maxwell's equation is used.

$\begin{matrix}{\begin{matrix}{{\int{\int_{D}{\nabla{\times E\;{dxdy}}}}} = {\oint_{\partial D}{E \cdot {ds}}}} \\{= {- {\partial_{t}{\int{\int_{D}{Bdxdy}}}}}}\end{matrix}\begin{matrix}{{\int{\int_{D}{\nabla{\times H\;{dxdy}}}}} = {\oint_{\partial D}{H \cdot {ds}}}} \\{= {\int{\int_{D}{\sigma\;{Edxdy}}}}}\end{matrix}} & \left( {6\text{-}1} \right)\end{matrix}$

D is two-dimensional area, ∂D is a counterclockwise boundary

If a rectangular area that is thin in the z-direction as the area D isdefined with the boundary surface interposed therebetween and doubleintegrals are made to approach 0 first, a condition that the tangentialcomponents E_(t) and H_(t) are continuous at the boundary surface isobtained. The electric field is not intended to be included in theboundary condition of the equation represented only by the magneticfield. However, in a case where the electrical conductivity is 0, it is∇×H=j=σE  (6-2)from the second expression of the expression (5-1), and it is impossibleto obtain the condition relating to the differential of H. Accordingly,air is virtually considered as a substance having a small but finiteelectrical conductivity, not a medium with electrical conductivity of 0and its electrical conductivity is assumed to be σ_(a). By doing this,the boundary condition that E_(t) is continuous is represented asfollows.

$\begin{matrix}{\left. \frac{\nabla{\times H}}{\sigma_{a}} \right|_{t^{air}} = \left. \frac{\nabla{\times H}}{\sigma_{c}} \right|_{t^{\begin{matrix}{battery} \\{electrode}\end{matrix}}}} & \left( {6\text{-}3} \right)\end{matrix}$This expression can be rewritten as follows.

$\begin{matrix}{\left. {\sigma_{a}^{- 1}\left( {\frac{\partial H_{z}}{\partial y} - \frac{\partial H_{y}}{\partial z}} \right)} \right|_{air} = {\left. {\sigma_{c}^{- 1}\left( {\frac{\partial H_{z}}{\partial y} - \frac{\partial H_{y}}{\partial z}} \right)} \middle| {}_{\begin{matrix}{battery} \\{electrode}\end{matrix}}{\sigma_{a}^{- 1}\left( {\frac{\partial H_{z}}{\partial x} - \frac{\partial H_{x}}{\partial z}} \right)} \right|_{air} = \left. {\sigma_{c}^{- 1}\left( {\frac{\partial H_{z}}{\partial x} - \frac{\partial H_{x}}{\partial z}} \right)} \right|_{\begin{matrix}{battery} \\{electrode}\end{matrix}}}} & \left( {6\text{-}4} \right)\end{matrix}$

The magnetic field is subjected to Fourier transformation as follows.

$\begin{matrix}{{\overset{\sim}{H}\left( {k_{x},k_{y},z,\omega} \right)} = {\int\limits_{- \infty}^{\infty}{\int\limits_{- \infty}^{\infty}{\int\limits_{- \infty}^{\infty}{e^{{{- i}\;\omega\; t} + {{ik}_{x}x} + {{ik}_{y}y}}{H\left( {x,y,z,t} \right)}{dxdydt}}}}}} & \left( {6\text{-}5} \right)\end{matrix}$From the expression (6-4), the following expression is obtained.

$\begin{matrix}{\left. {\sigma_{a}^{- 1}\left( {{{- {ik}_{y}}{\overset{\sim}{H}}_{z}} - \frac{\partial{\overset{\sim}{H}}_{y}}{\partial z}} \right)} \right|_{air} = {\left. {\sigma_{c}^{- 1}\left( {{{- {ik}_{y}}{\overset{\sim}{H}}_{z}} - \frac{\partial{\overset{\sim}{H}}_{y}}{\partial z}} \right)} \middle| {}_{\begin{matrix}{battery} \\{electrode}\end{matrix}}{\sigma_{a}^{- 1}\left( {{{- {ik}_{x}}{\overset{\sim}{H}}_{z}} - \frac{\partial{\overset{\sim}{H}}_{x}}{\partial z}} \right)} \right|_{air} = \left. {\sigma_{c}^{- 1}\left( {{{- {ik}_{x}}{\overset{\sim}{H}}_{z}} - \frac{\partial{\overset{\sim}{H}}_{x}}{\partial z}} \right)} \right|_{\begin{matrix}{battery} \\{electrode}\end{matrix}}}} & \left( {6\text{-}6} \right)\end{matrix}$

Next, consideration will be given on H_(n)(H_(z)). When it is assumedthat magnetic permeability is constant, there is the followingexpression relating to the magnetic field.div H=0  (6-7)Accordingly, H_(z) should be continuous at the boundary surface. Next,consideration will be given on the boundary condition for the zdifferential of H_(z). When the expression (6-7) is subjected to Fouriertransformation, the following expression is obtained.

$\begin{matrix}{{\frac{\partial}{\partial z}{\overset{\sim}{H}}_{z}} = {{{ik}_{x}{\overset{\sim}{H}}_{x}} + {{ik}_{y}{\overset{\sim}{H}}_{y}}}} & \left( {6\text{-}8} \right)\end{matrix}$

Since the tangential component H_(t) of the magnetic field iscontinuous, the right side is continuous at the boundary surface.Accordingly, the differential of the magnetic field H_(z) is continuousat the boundary surface.

$\begin{matrix}{\left. \frac{\partial{\overset{\sim}{H}}_{z}}{\partial z} \right|_{air} = \left. \frac{\partial{\overset{\sim}{H}}_{z}}{\partial z} \right|_{\begin{matrix}{battery} \\{electrode}\end{matrix}}} & \left( {6\text{-}9} \right)\end{matrix}$In this way, the boundary conditions were obtained. When these aresummarized once again, it is represented as follows.

$\begin{matrix}{\mspace{79mu}{\left. {\overset{\sim}{H}}_{x} \right|_{\begin{matrix}{battery} \\{electrode}\end{matrix}} = {\left. {\overset{\sim}{H}}_{x} \middle| {}_{air}\mspace{20mu}{\overset{\sim}{H}}_{y} \right|_{\begin{matrix}{battery} \\{electrode}\end{matrix}} = {\left. {\overset{\sim}{H}}_{y} \middle| {}_{air}\mspace{20mu}{\overset{\sim}{H}}_{z} \right|_{\begin{matrix}{battery} \\{electrode}\end{matrix}} = {\left. {\overset{\sim}{H}}_{z} \middle| {}_{air}{\sigma_{a}^{- 1}\left( {{{- {ik}_{y}}{\overset{\sim}{H}}_{z}} - \frac{\partial{\overset{\sim}{H}}_{y}}{\partial z}} \right)} \right|_{air} = {\left. {\sigma_{c}^{- 1}\left( {{{- {ik}_{y}}{\overset{\sim}{H}}_{z}} - \frac{\partial{\overset{\sim}{H}}_{y}}{\partial z}} \right)} \middle| {}_{\begin{matrix}{battery} \\{electrode}\end{matrix}}\mspace{20mu}{\sigma_{a}^{- 1}\left( {{{- {ik}_{x}}{\overset{\sim}{H}}_{z}} - \frac{\partial{\overset{\sim}{H}}_{x}}{\partial z}} \right)} \right|_{air} = {\left. {\sigma_{c}^{- 1}\left( {{{- {ik}_{x}}{\overset{\sim}{H}}_{z}} - \frac{\partial{\overset{\sim}{H}}_{x}}{\partial z}} \right)} \middle| {}_{\begin{matrix}{battery} \\{electrode}\end{matrix}}\mspace{20mu}\frac{\partial{\overset{\sim}{H}}_{z}}{\partial z} \right|_{air} = \left. \frac{\partial{\overset{\sim}{H}}_{z}}{\partial z} \right|_{\begin{matrix}{battery} \\{electrode}\end{matrix}}}}}}}}} & \left( {6\text{-}10} \right)\end{matrix}$The expression (6-10) corresponds to the above expression (10).

Finally, description will be made on changing the equations in the air.Since it is assumed that air is a substance having a virtually non-zeroelectrical conductivity, consideration should be given on matters thatthe following expression is established.

$\begin{matrix}{{\Delta\; H} = {\sigma_{a}\mu\frac{\partial}{\partial t}H}} & \left( {6\text{-}11} \right)\end{matrix}$When this is subjected to the Fourier transformation, it becomes asfollows.

$\begin{matrix}{{\left( {\frac{d^{2}}{{dz}^{2}} - k_{x}^{2} - k_{y}^{2} - {i\;{\omega\sigma}_{a}\mu}} \right)\overset{\sim}{H}} = 0} & \left( {6\text{-}12} \right)\end{matrix}$The general solution of this equation is represented as follows.{tilde over (H)}(k _(x) ,k _(y) ,z,ω)=a(k _(x) ,k _(y),ω)e ^(sz) +b(k_(x) ,k _(y),ω)e ^(−sz)   (6-13)Here, s is as follows.

$\begin{matrix}{{s^{2} = {k_{x}^{2} + k_{y}^{2} + {i\;{\omega\sigma\mu}}}}{s = {\frac{\sqrt{k_{x}^{2} + k_{y}^{2} + \sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\omega^{2}\sigma^{2}\mu^{2}}}}}{\sqrt{2}} + {i\frac{\sqrt{{- k_{x}^{2}} - k_{y}^{2} + \sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\omega^{2}\sigma^{2}\mu^{2}}}}}{\sqrt{2}}}}}} & \left( {6\text{-}14} \right)\end{matrix}$The expression (6-13) corresponds to the above expression (1)

a and b are obtained by measuring H=(H_(x), H_(y), H_(z)) on two planesin the air, each of which has a constant z-coordinate. If the expression(6-13) is used, it is possible to calculate the magnetic field at theboundary surface and its differential required for the boundarycondition expression (6-8).

It can be seen from the expressions (5-26) and (6-10) that H_(z) can beobtained independently from other components. When the result is used,the equations relating to H_(x) and H_(y) are obtained from theexpression (5-26).

<III-3 Analysis Overview by Inverse Problem Inside Battery>

A target of the inverse problem is written once again. In this section,the macro differential within the battery is represented by theζinstead of z. A governing equation of a target phenomenon is as followsinside the battery.

$\begin{matrix}{{{\mu\frac{\partial}{\partial t}H_{x}} = {{\left\{ {{\frac{\left( {{g_{e}\sigma_{c}} + {g_{c}\sigma_{e}}} \right)}{\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}} \right)} + {\frac{\left( {g_{c} + g_{e}} \right)}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)}\frac{\partial^{2}}{\partial ϛ^{2}}}} \right\} H_{x}} + {\frac{g_{e}{g_{c}\left( {\sigma_{c} - \sigma_{e}} \right)}^{2}}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\frac{\partial}{{\partial x}{\partial ϛ}}H_{z}}}}{{\mu\frac{\partial}{\partial t}H_{y}} = {{\left\{ {{\frac{\left( {{g_{e}\sigma_{c}} + {g_{c}\sigma_{e}}} \right)}{\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}} \right)} + {\frac{\left( {g_{c} + g_{e}} \right)}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)}\frac{\partial^{2}}{\partial ϛ^{2}}}} \right\} H_{y}} + {\frac{g_{e}{g_{c}\left( {\sigma_{c} - \sigma_{e}} \right)}^{2}}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\frac{\partial^{2}}{{\partial y}{\partial ϛ}}H_{z}}}}\mspace{20mu}{{\mu\frac{\partial}{\partial t}H_{z}} = {\frac{\left( {g_{e} + g_{c}} \right)}{\left( {{g_{e}\sigma_{e}} + {g_{c}\sigma_{c}}} \right)}\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial ϛ^{2}}} \right)H_{z}}}} & \left( {7\text{-}1} \right)\end{matrix}$

Here, the macro differential such as the expression (5-22) isrepresented by theζinstead of z.

$\begin{matrix}{\frac{\partial}{\partial ϛ} = {\frac{{\sigma_{c}g_{c}} + {\sigma_{e}g_{e}}}{\sigma_{c}\left( {g_{c} + g_{e}} \right)}\frac{\partial}{\partial z}}} & \left( {7\text{-}2} \right)\end{matrix}$Here,Q _(a)(k _(x) ,k _(y),0,ω)is a known function. From the expression (1-16), the boundary conditionis as follows.

$\begin{matrix}{\mspace{79mu}{\left. {\overset{\sim}{H}}_{x} \right|_{\begin{matrix}{battery} \\{electrode}\end{matrix}} = {\left. {\overset{\sim}{H}}_{x} \middle| {}_{air}\mspace{20mu}{\overset{\sim}{H}}_{y} \right|_{\begin{matrix}{battery} \\{electrode}\end{matrix}} = {\left. {\overset{\sim}{H}}_{y} \middle| {}_{air}\mspace{20mu}{\overset{\sim}{H}}_{z} \right|_{\begin{matrix}{battery} \\{electrode}\end{matrix}} = {\left. {\overset{\sim}{H}}_{z} \middle| {}_{air}\frac{\partial{\overset{\sim}{H}}_{y}}{\partial ϛ} \right|_{\begin{matrix}{battery} \\{electrode}\end{matrix}} = {\left. {\frac{\sigma_{c}}{\sigma_{a}}\frac{{\sigma_{c}g_{c}} + {\sigma_{e}g_{e}}}{\sigma_{a}{\sigma_{a}\left( {g_{c} + g_{e}} \right)}}\left( {{{{ik}_{y}\left( {1 - \frac{\sigma_{a}}{\sigma_{c}}} \right)}{\overset{\sim}{H}}_{z}} + \frac{\partial{\overset{\sim}{H}}_{y}}{\partial z}} \right)} \middle| {}_{air}\frac{\partial{\overset{\sim}{H}}_{x}}{\partial ϛ} \right|_{\begin{matrix}{battery} \\{electrode}\end{matrix}} = {\left. {\frac{\sigma_{c}}{\sigma_{a}}\frac{{\sigma_{c}g_{c}} + {\sigma_{e}g_{e}}}{\sigma_{a}\left( {g_{c} + g_{e}} \right)}\left( {{{{ik}_{x}\left( {1 - \frac{\sigma_{a}}{\sigma_{c}}} \right)}{\overset{\sim}{H}}_{z}} + \frac{\partial{\overset{\sim}{H}}_{x}}{\partial z}} \right)} \middle| {}_{air}\mspace{20mu}\frac{\partial{\overset{\sim}{H}}_{z}}{\partial ϛ} \right|_{\begin{matrix}{battery} \\{electrode}\end{matrix}} = \left. {\frac{{\sigma_{c}g_{c}} + {\sigma_{e}g_{e}}}{\sigma_{c}\left( {g_{c} + g_{e}} \right)}\frac{\partial{\overset{\sim}{H}}_{z}}{\partial z}} \right|_{air}}}}}}}} & \left( {7\text{-}3} \right)\end{matrix}$

The Fourier transformation of the expression (7-1) is represented asfollows.

$\begin{matrix}{{{i\;{\omega\mu}\;{\overset{\sim}{H}}_{x}} = {{\left\{ {{{- \frac{\left( {{g_{e}\sigma_{c}} + {g_{c}\sigma_{e}}} \right)}{\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}}\left( {k_{x}^{2} + k_{y}^{2}} \right)} + {\frac{\left( {g_{c} + g_{e}} \right)}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)}\frac{\partial^{2}}{\partial ϛ^{2}}}} \right\}{\overset{\sim}{H}}_{x}} - {{ik}_{x}\frac{g_{e}{g_{c}\left( {\sigma_{c} - \sigma_{e}} \right)}^{2}}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\frac{\partial}{\partial ϛ}{\overset{\sim}{H}}_{z}}}}{{i\;{\omega\mu}\;{\overset{\sim}{H}}_{y}} = {{\left\{ {{{- \frac{\left( {{g_{e}\sigma_{c}} + {g_{c}\sigma_{e}}} \right)}{\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}}\left( {k_{x}^{2} + k_{y}^{2}} \right)} + {\frac{\left( {g_{c} + g_{e}} \right)}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)}\frac{\partial^{2}}{\partial ϛ^{2}}}} \right\} H_{y}} - {{ik}_{y}\frac{g_{e}{g_{c}\left( {\sigma_{c} - \sigma_{e}} \right)}^{2}}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\frac{\partial}{\partial ϛ}H_{z}}}}\mspace{20mu}{{i\;{\omega\mu}\;{\overset{\sim}{H}}_{z}} = {\frac{\left( {g_{e} + g_{c}} \right)}{\left( {{g_{e}\sigma_{e}} + {g_{c}\sigma_{c}}} \right)}\left( {{- k_{x}^{2}} - k_{y}^{2} + \frac{\partial^{2}}{\partial ϛ^{2}}} \right){\overset{\sim}{H}}_{z}}}} & \left( {7\text{-}4} \right)\end{matrix}$Here, the ordinary differential equation relating to theζcan be solved using the boundary condition expression (7-3). Even in thecase of a three-dimensional magnetic field, it can be considered as inthe case of a two-dimensional magnetic field. However, since there is aterm of Hz as a difference, it is included as a special solution. Then,the general solution is set in a form including two unknown functions tosatisfy the boundary condition, and these unknown functions can bedetermined.

Second Embodiment

The configuration of the measurement device 10 according to a secondembodiment can be represented using FIG. 1, FIG. 3, and FIG. 4 similarlyto the measurement device 10 according to the first embodiment.

The measurement device 10 according to the present embodiment is thesame as the measurement device 10 according to the first embodimentexcept that the processing unit 160 generates three-dimensional magneticfield distribution information on the outside of the laminated body 20using only information indicating the distribution of the magnetic fieldon the first plane 201.

The acquisition unit 140 according to the present embodiment may measureonly the magnetic field of the first plane 201. Then, the processingunit 160 acquires in-plane distribution information includinginformation indicating the distribution of the magnetic field of thefirst plane 201 outside the laminated body 20. Here, the in-planedistribution information may not include information indicating thedistribution of the magnetic field of the second plane 202.

The processing unit 160 according to the present embodiment includes theoutside three-dimensional distribution generation unit 162 and theinside three-dimensional distribution generation unit 164 similarly asin the first embodiment. The outside three-dimensional distributiongeneration unit 162 generates three-dimensional magnetic fielddistribution information on the outside of the laminated body 20 byprocessing the in-plane distribution information.

In a case where the magnetic field (external magnetic field) from theouter side (side opposite to the laminated body 20) than the first plane201 which is the measurement plane of the magnetic field is sufficientlysmall, b=0 can be set in the expression (1) of the first embodiment. Inthat case, if a is obtained based on information indicating thedistribution of the magnetic field on one plane, the three-dimensionalmagnetic field distribution information on the outside of the laminatedbody 20 can be obtained. Therefore, the acquisition unit 140 does notneed to acquire information indicating the distribution of magneticfields on two planes.

Specifically, in the present embodiment, the three-dimensional magneticfield distribution information on the outside of the laminated body 20is represented by the following expression (12), and s is represented bythe following expression (2). Here, k_(x) is a wave number in thex-direction of the magnetic field, k_(y) is the wave number in they-direction of the magnetic field, z is a z-coordinate, ω is afrequency, σ is electrical conductivity, μ is magnetic permeability, andthe following expression (3) is a complex magnetic field vector. In thepresent embodiment, an outer space of the laminated body 20 is air, σ isthe electrical conductivity of air, and μ is the magnetic permeabilityof air. Then, the outside three-dimensional distribution generation unit162 obtains vectors a(k_(x), k_(y), ω) in the equation (12) usinginformation indicating the distribution of the magnetic field of thefirst plane 201.

$\begin{matrix}{{\overset{\sim}{H}\left( {k_{x},k_{y},z,\omega} \right)} = {{a\left( {k_{x},k_{y},\omega} \right)}e^{sz}}} & (12) \\{s = {\frac{\sqrt{k_{x}^{2} + k_{y}^{2} + \sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\omega^{2}\sigma^{2}\mu^{2}}}}}{\sqrt{2}} + {i\frac{\sqrt{{- k_{x}^{2}} - k_{y}^{2} + \sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\omega^{2}\sigma^{2}\mu^{2}}}}}{\sqrt{2}}}}} & (2) \\{H\left( {k_{x},k_{y},z,\omega} \right)} & (3)\end{matrix}$

That is, the outside three-dimensional distribution generation unit 162can process as follows. First, the outside three-dimensionaldistribution generation unit 162 acquires in-plane distributioninformation. Here, the in-plane distribution information includesinformation indicating the complex magnetic field vector at eachcoordinate as described in the first embodiment. It is assumed that thez-coordinate of the first plane 201 is z₁ and information indicating thedistribution of the magnetic field of the first plane 201 isH ₁(x,y,z ₁,ω).  (4)Then, the outside three-dimensional distribution generation unit 162obtains{tilde over (H)} ₁(k _(x) ,k _(y) ,z ₁,ω)  (6)by performing Fourier transformation on the information indicating thedistribution of the magnetic field of the first plane 201 with respectto x and y.

Then, the three-dimensional distribution of the complex magnetic fieldvector represented by the equation (12) is generated as thethree-dimensional magnetic field distribution information on the outsideof the laminated body 20. Here, a in the equation (12) is a vectorsatisfying the following expression (13).{tilde over (H)} ₁(k _(x) ,k _(y) ,z,ω)=a(k _(x) ,k _(y),ω)e ^(sz) ¹  (13)

In this manner, the three-dimensional magnetic field distributioninformation on the outside of the laminated body 20 is generated fromthe in-plane distribution information including information indicatingthe response characteristic to the change in the current applied by thecurrent applying unit 120. The three-dimensional magnetic fielddistribution information on the outside of the laminated body 20includes the frequency characteristic of the magnetic field. Thethree-dimensional magnetic field distribution information on the outsideof the laminated body 20 is data represented by, for example, amathematical expression or a table.

The inside three-dimensional distribution generation unit 164 processesthe three-dimensional magnetic field distribution information on theoutside of the laminated body 20 generated by the outsidethree-dimensional distribution generation unit 162 in the same manner asdescribed in the first embodiment to generate the three-dimensionalmagnetic field distribution information on the inside of the laminatedbody 20 as information on the inside of the laminated body 20.

The processing unit 160 may switch the processing method according tothe first embodiment and the processing method according to the secondembodiment according to the operation of the user with respect to themeasurement device 10.

Next, operations and effects of the present embodiment will bedescribed. In the present embodiment, similarly as in the firstembodiment, information on the inside of the laminated body 20 such as abattery can be obtained. Furthermore, it provides useful information fornon-destructive failure analysis and the like.

In the present embodiment, it is possible to know the three-dimensionalmagnetic field distribution inside the laminated body 20 such as abattery. Therefore, it is possible to specifically know the situationinside the laminated body 20.

In addition, since the processing unit 160 can generate thethree-dimensional magnetic field distribution information on the outsideof the laminated body 20 using only information illustrating thedistribution of the magnetic field on the first plane 201, the currentapplying unit 120 does not need to measure the magnetic field of thesecond plane 202. Therefore, measurement time can be shortened.

Third Embodiment

FIG. 5 is a block diagram illustrating a functional configuration of theprocessing unit 160 according to the third embodiment. The measurementdevice 10 according to the present embodiment is the same as themeasurement device 10 according to the first embodiment or the secondembodiment except that the processing unit 160 specifies the position ofthe defect inside the laminated body 20 based on the three-dimensionalmagnetic field distribution information on the outside of the laminatedbody 20.

The processing unit 160 according to the present embodiment includes theoutside three-dimensional distribution generation unit 162 and thedefect specifying unit 166. The outside three-dimensional distributiongeneration unit 162 processes the in-plane distribution informationacquired by the acquisition unit 140 to generate the three-dimensionalmagnetic field distribution information on the outside of the laminatedbody 20. The contents of processing performed by the outsidethree-dimensional distribution generation unit 162 are the same as thoseof the measurement device 10 according to the first embodiment or thesecond embodiment. Then, the defect specifying unit 166 processes thethree-dimensional magnetic field distribution information on the outsideof the laminated body 20 generated by the outside three-dimensionaldistribution generation unit 162 to specify the position of the defectinside the laminated body 20 as the information inside the laminatedbody 20.

More specifically, based on the three-dimensional magnetic fielddistribution information on the outside of the laminated body 20, theboundary condition between the magnetic fields inside and outside thelaminated body 20, and the equation on the magnetic field inside thelaminated body 20, the defect specifying unit 166 specifies the positionof the defect inside the laminated body 20. Here, the defect is a defectin which a magnetic field of some magnitude is localized.

The boundary condition between the magnetic fields inside and outsidethe laminated body 20 is represented by the expression (10)

$\begin{matrix}{\mspace{79mu}{\left. {\overset{\sim}{H}}_{x} \right|_{inside} = {\left. {\overset{\sim}{H}}_{x} \middle| {}_{outside}\mspace{20mu}{\overset{\sim}{H}}_{y} \right|_{inside} = {\left. {\overset{\sim}{H}}_{y} \middle| {}_{outside}\mspace{20mu}{\overset{\sim}{H}}_{z} \right|_{inside} = {\left. H_{z} \middle| {}_{outside}{\sigma_{a}^{- 1}\left( {{{- {ik}_{y}}{\overset{\sim}{H}}_{z}} - \frac{\partial{\overset{\sim}{H}}_{y}}{\partial z}} \right)} \right|_{outside} = {\left. {\sigma_{c}^{- 1}\left( {{{- {ik}_{y}}{\overset{\sim}{H}}_{z}} - \frac{\partial{\overset{\sim}{H}}_{y}}{\partial z}} \right)} \middle| {}_{inside}\mspace{20mu}{\sigma_{a}^{- 1}\left( {{{- {ik}_{x}}{\overset{\sim}{H}}_{z}} - \frac{\partial{\overset{\sim}{H}}_{x}}{\partial z}} \right)} \right|_{outside} = {\left. {\sigma_{c}^{- 1}\left( {{{- {ik}_{x}}{\overset{\sim}{H}}_{z}} - \frac{\partial{\overset{\sim}{H}}_{x}}{\partial z}} \right)} \middle| {}_{inside}\mspace{20mu}\frac{\partial{\overset{\sim}{H}}_{z}}{\partial z} \right|_{outside} = \left. \frac{\partial{\overset{\sim}{H}}_{z}}{\partial z} \right|_{inside}}}}}}}} & (10)\end{matrix}$

Here,{tilde over (H)} _(x)is the x-component of the complex magnetic field vector,{tilde over (H)} _(y)is the y-component of the complex magnetic field vector,{tilde over (H)} _(z)is the z-component of the complex magnetic field vector, σ_(a) is theelectrical conductivity of outside the laminated body 20, and σ_(c) isthe electrical conductivity of the first layer 210. Here, the firstlayer 210 is the outermost layer on the first plane 201 side of thelaminated body 20.

In addition, an equation relating to the magnetic field inside thelaminated body 20 (time reverse equation) is specifically represented bythe following expression (14). Here, μ is magnetic permeability insideof the laminated body 20, t is time, H_(x) is a component in thex-direction of the magnetic field, H_(y) is a component in they-direction of the magnetic field, H_(z) is a component in thez-direction of the magnetic field, g_(c) is a thickness of the firstlayer 210, g_(e) is a thickness of the second layer 220, σ_(c) is theelectrical conductivity of the first layer 210, and σ_(e) is theelectrical conductivity of the second layer 220. The magneticpermeability μ can be regarded as uniform in the outside of thelaminated body 20 and inside the laminated body 20.

$\begin{matrix}{{{- \mu}\frac{\partial}{\partial t}H_{x}} = {{{\left\{ {{\frac{\left( {{g_{e}\sigma_{c}} + {g_{c}\sigma_{e}}} \right)}{\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}} \right)} + {\frac{\left( {g_{c} + g_{e}} \right)}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)}\frac{\partial^{2}}{\partial z^{2}}}} \right\} H_{x}} + {\frac{g_{e}{g_{c}\left( {\sigma_{c} - \sigma_{e}} \right)}^{2}}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\frac{\partial}{{\partial x}{\partial z}}H_{z}} - {\mu\frac{\partial}{\partial t}H_{y}}} = {{{\left\{ {{\frac{\left( {{g_{e}\sigma_{c}} + {g_{c}\sigma_{e}}} \right)}{\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}} \right)} + {\frac{\left( {g_{c} + g_{e}} \right)}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)}\frac{\partial^{2}}{\partial z^{2}}}} \right\} H_{y}} + {\frac{g_{e}{g_{c}\left( {\sigma_{c} - \sigma_{e}} \right)}^{2}}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\frac{\partial^{2}}{{\partial y}{\partial z}}H_{z}}\mspace{20mu} - {\mu\frac{\partial}{\partial t}H_{z}}} = {\frac{\left( {g_{e} + g_{c}} \right)}{\left( {{g_{e}\sigma_{e}} + {g_{c}\sigma_{c}}} \right)}\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}}} \right)H_{z}}}}} & (14)\end{matrix}$

The defect specifying unit 166 derives H_(x), H_(y), and H_(z) thatsatisfy the expression (14) based on the three-dimensional magneticfield distribution information on the outside of the laminated body 20and the boundary condition between a magnetic field inside and outsideof the laminated body 20, and specifies the coordinates at which H_(x),H_(y), and H_(z) diverge as the position of the defect. Specifically, inthe boundary condition, the magnetic field outside the laminated body 20and the differential thereof are derived from the expression (1) of thefirst embodiment or the expression (12) of the second embodiment. Then,a solution of the expression (14) which is a time reverse equation isobtained. In a case where the coordinates at which the obtained H_(x),H_(y), and H_(z) diverge respectively appear, the coordinates arespecified as (x, y, z) coordinates of the defect. Here, the coordinatesat which H_(x), H_(y), and H_(z) diverge respectively refer tocoordinates at which peaks appear specifically for at least one ofH_(x), H_(y), and H_(z) on data. Each mathematical expression will bedescribed later in detail.

Information which indicates the position (coordinates) of the defect andis output from the defect specifying unit 166 is input to the displayunit 180, for example. Then, on the display unit 180, data fordisplaying the position of the defect in the laminated body 20 isgenerated as an image. Then, it is possible to cause the monitor 390 todisplay the defect position of the laminated body 20 as an image. Inaddition, the display unit 180 may output the coordinates of the defectby displaying characters.

The processing unit 160 may output the (x, y, z) coordinates of thedefect or may output only the z-coordinate of the defect.

Next, operations and effects of the present embodiment will bedescribed. In the present embodiment, similarly as in the firstembodiment and the second embodiment, information on the inside of thelaminated body 20 such as a battery can be obtained. Furthermore, itprovides useful information for non-destructive failure analysis and thelike.

In the present embodiment, it is possible to specify the position of aspecific defect.

In the following, analysis of a defect model of the laminated body 20will be described. In the following description, the laminated body 20is described as a battery, but the laminated body 20 is not limited tothe battery.

<IV 3D Inverse Analysis Simulation of Magnetic Field>

<IV-1 Defect Model>

The expression (5-26) is as follows. Here, a macro variable is writtenas z.

$\begin{matrix}{{{\mu\frac{\partial}{\partial t}H_{x}} = {{\left\{ {{\frac{\left( {{g_{e}\sigma_{c}} + {g_{c}\sigma_{e}}} \right)}{\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}} \right)} + {\frac{\left( {g_{c} + g_{e}} \right)}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)}\frac{\partial^{2}}{\partial z^{2}}}} \right\} H_{x}} + {\frac{g_{e}{g_{c}\left( {\sigma_{c} - \sigma_{e}} \right)}^{2}}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\frac{\partial^{2}}{{\partial x}{\partial z}}H_{z}}}}{{\mu\frac{\partial}{\partial t}H_{y}} = {{\left\{ {{\frac{\left( {{g_{e}\sigma_{c}} + {g_{c}\sigma_{e}}} \right)}{\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}} \right)} + {\frac{\left( {g_{c} + g_{e}} \right)}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)}\frac{\partial^{2}}{\partial z^{2}}}} \right\} H_{y}} + {\frac{g_{e}{g_{c}\left( {\sigma_{c} - \sigma_{e}} \right)}^{2}}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\frac{\partial^{2}}{{\partial y}{\partial z}}H_{z}}}}\mspace{20mu}{{\mu\frac{\partial}{\partial t}H_{z}} = {\frac{\left( {g_{e} + g_{c}} \right)}{\left( {{g_{e}\sigma_{e}} + {g_{c}\sigma_{c}}} \right)}\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}}} \right)H_{z}}}} & \left( {8\text{-}1} \right)\end{matrix}$

Here, only H_(z) satisfies an independent equation. When H_(z) isdecided, H_(x), and H_(y) can be solved by making it a forced term.Accordingly, the equation relating only to H_(z) should be considered atfirst.

$\begin{matrix}{{\mu\frac{\partial}{\partial t}H_{z}} = {\frac{\left( {g_{e} + g_{c}} \right)}{\left( {{g_{e}\sigma_{e}} + {g_{c}\sigma_{c}}} \right)}\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}}} \right)H_{z}}} & \left( {8\text{-}2} \right)\end{matrix}$

The coefficient is set as follows for simplicity.

$\begin{matrix}{\lambda = \frac{g_{c} + g_{e}}{\mu\left( {{\sigma_{c}g_{c}} + {\sigma_{e}g_{e}}} \right)}} & \left( {8\text{-}3} \right)\end{matrix}$In this case, the expression (8-1) becomes as follows.

$\begin{matrix}{{\frac{\partial}{\partial t}\Phi} = {{\lambda\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}}} \right)}\Phi}} & \left( {8\text{-}4} \right)\end{matrix}$

This equation represents a state in which no magnetic field is generatedinside the battery. It is The model when there is a defect is assumed asfollows with the analogy of the problem of heat conduction. A model inwhich defects whose magnitude is proportional to Φ_(L) at space time(t=0, x=x₀, y=y₀, z=z₀) are localized and the magnetic field leaks isconsidered.

$\begin{matrix}{\frac{\partial}{\partial\Phi} = {{{\lambda\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}}} \right)}\Phi} + {\Phi_{L}{\delta\left( {x - x_{0}} \right)}{\delta\left( {y - y_{0}} \right)}{\delta\left( {z - z_{0}} \right)}}}} & \left( {8\text{-}5} \right)\end{matrix}$

Φ is subjected to the Fourier transformation as follows.

${\overset{\sim}{\Phi}\left( {k_{x},k_{y},z,\omega} \right)} = {\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{e^{{{- i}\;\omega\; t} + {{ik}_{x}x} + {{ik}_{y}y}}{\Phi\left( {x,y,z,t} \right)}{dxdydt}}}}}$From the expression (8-5), the following expression is obtained.

$\begin{matrix}{{i\;\omega\;\overset{\sim}{\Phi}} = {{{\lambda\left( {\frac{\partial^{2}}{\partial z^{2}} - k_{x}^{2} - k_{y}^{2}} \right)}\overset{\sim}{\Phi}} + {\Phi_{L}e^{{{ik}_{x}x_{0}} + {{ik}_{y}y_{0}}}{\delta\left( {z - z_{0}} \right)}}}} & \left( {8\text{-}6} \right)\end{matrix}$This expression is transformed to obtain the following expression.

$\begin{matrix}{{{\lambda\;\frac{\partial^{2}}{\partial z^{2}}\overset{\sim}{\Phi}} - {\left\{ {{\lambda\left( {k_{x}^{2} + k_{y}^{2}} \right)} + {i\;\omega}} \right\}\overset{\sim}{\Phi}}} = {{- \Phi_{L}}e^{{{ik}_{x}x_{0}} + {{ik}_{y}y_{0}}}{\delta\left( {z - z_{0}} \right)}}} & \left( {8\text{-}7} \right)\end{matrix}$

The following Green function G₀(z, z₀, k) is considered.

$\begin{matrix}{{G_{0}\left( {z,z_{0},s} \right)} = {{{\frac{1}{2s}e^{{- s}{{z - z_{0}}}}\frac{\partial^{2}}{\partial z^{2}}{G_{0}\left( {z,z_{0},s} \right)}} - {s^{2}{G_{0}\left( {z,z_{0},k} \right)}}} = {\delta\left( {z - z_{0}} \right)}}} & \left( {8\text{-}8} \right)\end{matrix}$The solution of the expression (8-7) is represented as follows.

$\begin{matrix}{\mspace{20mu}{{\overset{\sim}{\Phi} = {{{- \Phi_{L}}e^{{{ik}_{x}x_{0}} + {{ik}_{y}y_{0}}}{G_{0}\left( {z,z_{0},s} \right)}} + {c_{1}e^{sz}} + {c_{2}e^{- {sz}}}}}{s = {\sqrt{k_{x}^{2} + k_{y}^{2} + \frac{i\;\omega}{\lambda}} = {{\frac{1}{\sqrt{2}}\sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right) + \sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + \left( \frac{\omega}{\lambda} \right)^{2}}}} + {\frac{i}{\sqrt{2}}\sqrt{{- \left( {k_{x}^{2} + k_{y}^{2}} \right)} + \sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + \left( \frac{\omega}{\lambda} \right)^{2}}}}}}}\mspace{20mu}{{G_{0}\left( {z,z_{0},s} \right)} = {\frac{1}{2s}e^{{- s}{{z - z_{0}}}}}}}} & \left( {8\text{-}9} \right)\end{matrix}$

It is assumed that the differential coefficients{tilde over (Φ)}₀and{tilde over ({dot over (Φ)})}₀for Φ and its z on the surface of the battery are given. At the boundaryz=0, the following expression is established.

$\begin{matrix}{{{{{- \Phi_{L}}e^{{{ik}_{x}x_{0}} + {{ik}_{y}y_{0}}}{G_{0}\left( {0,z_{0},s} \right)}} + c_{1} + c_{2}} = {{{\overset{\sim}{\Phi}}_{0} - {\Phi_{L}e^{{{ik}_{x}x_{0}} + {{ik}_{y}y_{0}}}{G_{0}\left( {0,z_{0},s} \right)}} + c_{1} - c_{2}} = \frac{{\overset{\overset{.}{\sim}}{\Phi}}_{0}}{s}}}{{G_{0}\left( {0,z_{0},s} \right)} = {\frac{1}{2s}{e^{- {sz}_{0}}\left( {\because{0 < z_{0}}} \right)}}}} & \left( {8\text{-}10} \right)\end{matrix}$When this expression is solved for c₁ and c₂, the following expressionis obtained.

$\begin{matrix}{{c_{1} = {{\frac{1}{2}\left( {{\overset{\sim}{\Phi}}_{0} + \frac{{\overset{\overset{.}{\sim}}{\Phi}}_{0}}{s}} \right)} + {\Phi_{L}e^{{{ik}_{x}x_{0}} + {{ik}_{y}y_{0}}}\frac{1}{2s}e^{{sz}_{0}}}}}{c_{2} = {\frac{1}{2}\left( {{\overset{\sim}{\Phi}}_{0} - \frac{{\overset{\overset{.}{\sim}}{\Phi}}_{0}}{s}} \right)}}} & \left( {8\text{-}11} \right)\end{matrix}$

A general solution of the equation (8-5) is obtained as in theexpressions (8-9) and (8-11).

<IV-2 Inverse Analysis>

Given a boundary condition (assuming that c₁=0, c₂=0, that is, there isno magnetic field other than the leak magnetic field)

$\begin{matrix}{{{\overset{\sim}{\Phi}}_{0} = {{- \Phi_{L}}e^{{{ik}_{x}x_{0}} + {{ik}_{y}y_{0}}}{G_{0}\left( {0,z_{0},s} \right)}}}{{\overset{\overset{.}{\sim}}{\Phi}}_{0} = {{- s}\;\Phi_{L}e^{{{ik}_{x}x_{0}} + {{ik}_{y}y_{0}}}{G_{0}\left( {0,z_{0},s} \right)}}}{{G_{0}\left( {0,z_{0},s} \right)} = {\frac{1}{2s}{e^{- {sz}_{0}}\left( {0 < z_{0}} \right)}}}} & \left( {9\text{-}1} \right)\end{matrix}$with z=0, and consideration is given on the problem of inverselyobtaining the second termΦ_(L)δ(t)δ(x−x ₀)δ(y−y ₀)δ(z−z ₀)  (9-2)on the right side of the expression (8-4). The equation used for theinverse analysis is the time reverse equation.

$\begin{matrix}{{{- \frac{\partial}{\partial t}}\Phi} = {{\lambda\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}}} \right)}\Phi}} & \left( {9\text{-}3} \right)\end{matrix}$That is, from the measurement result at t>0, the leak magnetic fieldρ(x, y, z) at t=0 is estimated as follows.

$\begin{matrix}{{\rho\left( {x,y,z} \right)} = {\lim\limits_{t->{+ 0}}{\Phi\left( {x,y,z,t} \right)}}} & \left( {9\text{-}4} \right)\end{matrix}$

The solution of the equation (9-3) given the boundary condition iseasily obtained and is as follows.

$\begin{matrix}{{\overset{\sim}{\Phi} = {{c_{1}e^{kz}} + {c_{2}e^{{- {kz}}\;}}}}{c_{1} = {\frac{1}{2}\left( {{\overset{\sim}{\Phi}}_{0} + \frac{{\overset{\overset{.}{\sim}}{\Phi}}_{0}}{k}} \right)}}{c_{2\;} = {\frac{1}{2}\left( {{\overset{\sim}{\Phi}}_{0} - \frac{{\overset{\overset{.}{\sim}}{\Phi}}_{0}}{k}} \right)}}} & \left( {9\text{-}5} \right)\end{matrix}$Here, k is a complex conjugate of s and is given by the followingexpression.

$\begin{matrix}{\mspace{20mu}{{\sigma = \frac{{g_{e}\sigma_{e}} + {g_{c}\sigma_{c}}}{g_{e} + g_{c}}}\mspace{20mu}{\lambda = {\frac{g_{c} + g_{e}}{\mu\left( {{\sigma_{c}g_{c}} + {\sigma_{e}g_{e}}} \right)} = \frac{1}{\mu\sigma}}}}} & \left( {9\text{-}6} \right) \\{k = {\sqrt{k_{x}^{2} + k_{y}^{2} - {i\;\mu\;\sigma\;\omega}} = {{\frac{1}{\sqrt{2}}\sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right) + \sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\mu^{2}\sigma^{2}\omega^{2}}}}} - {\frac{i}{\sqrt{2}}\sqrt{{- \left( {k_{x}^{2} + k_{y}^{2}} \right)} + \sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\mu^{2}\sigma^{2}\omega^{2}}}}}}}} & \left( {9\text{-}7} \right)\end{matrix}$This is subjected to inverse Fourier transformation to obtain Φ(x, y, z,t) of the expression (9-4).

$\begin{matrix}{{\Phi\left( {x,y,t} \right)} = {\left( \frac{1}{2\pi} \right)^{3}{\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{\int_{0}^{\infty}{e^{{i\;\omega\; t} - {{ik}_{x}x} - {{ik}_{y}y}}{\overset{\sim}{\Phi}\left( {k_{x},k_{y},z,\omega} \right)}{dk}_{x}{dk}_{y}d\;\omega}}}}}} & \left( {9\text{-}8} \right)\end{matrix}$

<IV-2-1 Case of One-Dimensional k_(x)=k_(y)=0>

The following time reverse equation is solved.

$\begin{matrix}{{{- \frac{\partial}{\partial t}}\Phi} + {\lambda\;\frac{\partial^{2}}{\partial z^{2}}\Phi}} & \left( {9\text{-}9} \right)\end{matrix}$Average electrical conductivity is defined as follows.

$\begin{matrix}{\sigma = \frac{{g_{e}\sigma_{e}} + {g_{c}\sigma_{c}}}{g_{e} + g_{c}}} & \left( {9\text{-}10} \right)\end{matrix}$s and k can be written as follows.

$\begin{matrix}{{s = {{\frac{1 + i}{\sqrt{2}}\sqrt{\frac{\omega}{\lambda}}} = {\frac{1 + i}{\sqrt{2}}\sqrt{\mu\;\sigma\;\omega}}}}{k = {{\frac{\left( {1 - i} \right)}{\sqrt{2}}\sqrt{\frac{\omega}{\lambda}}} = {\frac{\left( {1 - i} \right)}{\sqrt{2\;}}\sqrt{\mu\;\sigma\;\omega}}}}} & \left( {9\text{-}11} \right)\end{matrix}$

A defect in the form of the right side of expression (8-5) is set atarget. In this case, a magnetic field as in the expression (9-1) isgenerated at the boundary. At the boundary z=0, the boundary conditionis assumed as follows.

$\begin{matrix}{{{\overset{\sim}{\Phi}}_{0} = {{- \Phi_{L}}\frac{1}{2s}e^{- {sz}_{0}}}}{{\overset{\overset{.}{\sim}}{\Phi}}_{0} = {{- \Phi_{L}}\frac{1}{2}e^{- {sz}_{0}}}}} & \left( {9\text{-}12} \right)\end{matrix}$The solution of the expression (9-9) is as follows from the expression(9-5).

$\begin{matrix}{{{\overset{\sim}{\Phi} = {{c_{1}e^{kz}} + {c_{2}e^{- {kz}}}}}c_{1\;} = {\frac{1}{2}\left( {{\overset{\sim}{\Phi}}_{0} + \frac{{\overset{\overset{.}{\sim}}{\Phi}}_{0}}{k}} \right)}}{c_{2} = {\frac{1}{2}\left( {{\overset{\sim}{\Phi}}_{0} - \frac{{\overset{\overset{.}{\sim}}{\Phi}}_{0}}{k}} \right)}}} & \left( {9\text{-}13} \right)\end{matrix}$c₁ and c₂ are as follows.

$\begin{matrix}{{c_{1\;} = {\frac{- \Phi_{L}}{2\sqrt{2}}\frac{1}{\sqrt{\mu\;\sigma\;\omega}}e^{- {sz}_{0}}}}{c_{2} = {\frac{i\;\Phi_{L}}{2\sqrt{2}}\frac{1}{\sqrt{\mu\;\sigma\;\omega}}e^{- {sz}_{0}}}}} & \left( {9\text{-}14} \right)\end{matrix}$

The expression (9-13) is as follows.

$\begin{matrix}\begin{matrix}{\overset{\sim}{\Phi} = {{c_{1}e^{kz}} + {c_{2}e^{- {kz}}}}} \\{= {\frac{\Phi_{L}}{2\sqrt{2}}\sqrt{\frac{\lambda}{\omega}}\left( {{- e^{{- {sz}_{0}} + {kz}}} + {ie}^{{- {sz}_{0}} - {kz}}} \right)}} \\{= {\frac{\Phi_{L}}{2\sqrt{2}}{\sqrt{\frac{1}{\mu\;{\sigma\omega}}}\left\lbrack {{- e^{\sqrt{\frac{\omega\;\sigma\;\omega}{2}}{\{{{{- {({1 + i})}}z_{0}} + {{({1 - i})}z}}\}}}} + {ie}^{\sqrt{\frac{\mu\;\sigma\;\omega}{2}}{\{{{{- {({1 + i})}}z_{0}} - {{({1 - i})}z}}\}}}} \right\rbrack}}}\end{matrix} & \left( {9\text{-}15} \right)\end{matrix}$The result of inverse analysis is as follows.

$\begin{matrix}{{\overset{\sim}{\Phi}\left( {z,t} \right)} = {\frac{\Phi_{L}}{2\sqrt{2}}\sqrt{\frac{1}{\mu\;\sigma}}{\int_{0}^{\infty}{\frac{{- e^{\sqrt{\frac{{\mu\sigma}\;\omega}{2}}{\{{{{- {({1 + i})}}z_{0}} + {{({1 - i})}z}}\}}}} + {ie}^{\sqrt{\frac{\mu\;\sigma\;\omega}{2}}{\{{{{- {({1 + i})}}z_{0}} - {{({1 - i})}z}}\}}}}{\sqrt{\omega}}e^{i\;\omega\; t}d\;\omega}}}} & \left( {9\text{-}16} \right)\end{matrix}$

In the specific calculation of inverse analysis, the following ingenuityis necessary.

$\begin{matrix}{{\Phi\left( {z,t} \right)} = {\frac{\Phi_{L}}{2\sqrt{2}}\sqrt{\frac{1}{\mu\;\sigma}}{\int_{0}^{\infty}{\frac{{- e^{\sqrt{\frac{{\mu\sigma}\;\omega}{2}}{\{{{- {({1 + i})}}{z_{0}{({\alpha - i})}}z}\}}}} + {ie}^{\sqrt{\frac{\mu\;\sigma\;\omega}{2}}{\{{{{- {({1 + i})}}z_{0}} - {{({\alpha - i})}z}}\}}}}{\sqrt{\omega}}e^{i\;\omega\; t}d\;\omega}}}} & \left( {9\text{-}17} \right)\end{matrix}$In inverse analysis, when it is set that α<<1, it becomes goodconvergence

<IV-2-2 Three Dimension>

Solve the following time reverse equation.

$\begin{matrix}{{{- \frac{\partial}{\partial t}}\Phi} = {{\lambda\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}}} \right)}\Phi}} & \left( {9\text{-}18} \right)\end{matrix}$The average electrical conductivity is defined as follows.

$\begin{matrix}{\sigma = \frac{{g_{e}\sigma_{e}} + {g_{c}\sigma_{c}}}{g_{e} + g_{c}}} & \left( {9\text{-}19} \right)\end{matrix}$s and k can be written as follows.

$\begin{matrix}{{s = {\sqrt{k_{x}^{2} + k_{y}^{2} - {i\;\mu\;\sigma\;\omega}} = {{\frac{1}{\sqrt{2}}\sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right) + \sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\mu^{2}\sigma^{2}\omega^{2}}}}} + {\frac{i}{\sqrt{2}}\sqrt{{- \left( {k_{x}^{2} + k_{y}^{2}} \right)} + \sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\mu^{2}\sigma^{2}\omega^{2}}}}}}}}{k = {\sqrt{k_{x}^{2} + k_{y}^{2} - {i\;\mu\;{\sigma\omega}}} = {{\frac{1}{\sqrt{2}}\sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right) + \sqrt{{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\mu^{2}\sigma^{2}\omega^{2}}}\;}}} - {\frac{i}{\sqrt{2}}\sqrt{{- \left( {k_{x}^{2} + k_{y}^{2}} \right)} + \sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\mu^{2}\sigma^{2}\omega^{2}}}}}}}}} & \left( {9\text{-}20} \right)\end{matrix}$

The defect in the form of the right side of the expression (8-5) is seta target. In this case, a magnetic field as in the expression (9-1) isgenerated at the boundary. At the boundary z=0, the boundary conditionis assumed as follows.

$\begin{matrix}{{{\overset{\sim}{\Phi}}_{0} = {{- \frac{\Phi_{L}}{2s}}e^{{{ik}_{x}x_{0}} + {{ik}_{y}y_{0}}}e^{- {sz}_{0}}}}{{\overset{\overset{.}{\sim}}{\Phi}}_{0} = {{- \frac{\Phi_{L}}{2}}e^{{{ik}_{x}x_{0}} + {{ik}_{y}y_{0}}}e^{- {sz}_{0}}}}} & \left( {9\text{-}21} \right)\end{matrix}$The solution of the expression (9-9) is as follows from the expression(9-5).

$\begin{matrix}{{{\overset{\sim}{\Phi} = {{c_{1}e^{kz}} + {c_{2}e^{- {kz}}}}}c_{1} = {\frac{1}{2}\left( {{\overset{\sim}{\Phi}}_{0} + \frac{{\overset{\overset{.}{\sim}}{\Phi}}_{0}}{k}} \right)}}{c_{2} = {\frac{1}{2}\left( {{\overset{\sim}{\Phi}}_{0} - \frac{{\overset{\overset{.}{\sim}}{\Phi}}_{0}}{k}} \right)}}} & \left( {9\text{-}22} \right)\end{matrix}$The c₁ and c₂ are as follows.

$\begin{matrix}{{c_{1} = {{- \Phi_{L}}\frac{\sqrt{2}}{4}e^{{{ik}_{x}x_{0}} + {{ik}_{y}y_{0}}}e^{- {sz}_{0}}\frac{\sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right) + \sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\mu^{2}\sigma^{2}\omega^{2}}}}}{\sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\mu^{2}\sigma^{2}\omega^{2}}}}}}{c_{2} = {i\;\Phi_{L}\frac{\sqrt{2}}{4}e^{{{ik}_{x}x_{0}} + {{ik}_{y}y_{0}}}e^{- {sz}_{0}}\frac{\sqrt{{- \left( {k_{x}^{2} + k_{y}^{2}} \right)} + \sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\mu^{2}\sigma^{2}\omega^{2}}}}}{\sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\mu^{2}\sigma^{2}\omega^{2}}}}}}} & \left( {9\text{-}23} \right)\end{matrix}$

The result of inverse analysis is as follows.

$\begin{matrix}{\overset{\sim}{\Phi} = {{{- \Phi_{L}}\frac{\sqrt{2}}{4}e^{{{ik}_{x}x_{0}} + {{ik}_{y}y_{0}}}e^{{kz} - {sz}_{0}}\frac{\sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right) + \sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\mu^{2}\sigma^{2}\omega^{2}}}}}{\sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\mu^{2}\sigma^{2}\omega^{2}}}}} + {i\;\Phi_{L}\frac{\sqrt{2}}{4}e^{{{ik}_{x}x_{0}} + {{ik}_{y}y_{0}}}e^{{- {sz}_{0}} - {kz}}\frac{\sqrt{{- \left( {k_{x}^{2} + k_{y}^{2}} \right)} + \sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\mu^{2}\sigma^{2}\omega^{2}}}}}{\sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\mu^{2}\sigma^{2}\omega^{2}}}}}}} & \left( {9\text{-}24} \right)\end{matrix}$This is subjected to the inverse Fourier transformation to obtain thefollowing expression.

$\begin{matrix}\begin{matrix}{{\Phi\left( {x,y,z,t} \right)} = {{\Phi_{L}\left( \frac{1}{2\pi} \right)}^{3}{\int\limits_{- \infty}^{\infty}{\int\limits_{- \infty}^{\infty}{\int\limits_{0}^{\infty}e^{{i\;\omega\; t} - {ik}_{x} - x - {{ik}_{y}y}}}}}}} \\{{\overset{\sim}{\Phi}\left( {k_{x},k_{y},z,\omega} \right)}{dk}_{x}{dk}_{y}d\;\omega} \\{= {{\Phi_{L}\left( \frac{1}{2\pi} \right)}^{3}{\int\limits_{- \infty}^{\infty}{\int\limits_{- \infty}^{\infty}{\int\limits_{0}^{\infty}e^{{i\;\omega\; t} - {{ik}_{x}{({x - x_{0}})}} - {{ik}_{y}{({y - y_{0}})}}}}}}}} \\{\begin{Bmatrix}{{{- \frac{\sqrt{2}}{4}}e^{{kz} - {sz}_{0}}\frac{\sqrt{\begin{matrix}{\left( {k_{x}^{2} + k_{y}^{2}} \right) +} \\\sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\mu^{2}\sigma^{2}\omega^{2}}}\end{matrix}}}{\sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\mu^{2}\sigma^{2}\omega^{2}}}}} +} \\{i\;\frac{\sqrt{2}}{4}e^{{- {kz}} - {sz}_{0}}\frac{\sqrt{\begin{matrix}{{- \left( {k_{x}^{2} + k_{y}^{2}} \right)} +} \\\sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\mu^{2}\sigma^{2}\omega^{2}}}\end{matrix}}}{\sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\mu^{2}\sigma^{2}\omega^{2}}}}}\end{Bmatrix}} \\{{dk}_{x}{dk}_{y}d\;\omega}\end{matrix} & \left( {9\text{-}25} \right)\end{matrix}$As described above, the behavior of the solution of the time reverseequation in the case of having a defect can be understood. The aboveexpression (14) illustrates the time reverse equation for each elementof x, y, and z.

Fourth Embodiment

The configuration of the measurement device 10 according to the fourthembodiment can be described with reference to FIG. 5 similarly to themeasurement device 10 according to the third embodiment. The measurementdevice 10 according to the fourth embodiment is the same as themeasurement device 10 according to the third embodiment except for thepoints to be described below. The measurement device 10 according to thefourth embodiment specifies only the depth at which the defect occurs inthe laminated body 20.

The processing unit 160 according to the present embodiment includes theoutside three-dimensional distribution generation unit 162 and thedefect specifying unit 166 similarly as in the third embodiment. Theoutside three-dimensional distribution generation unit 162 processes thein-plane distribution information acquired by the acquisition unit 140to generate the three-dimensional magnetic field distributioninformation on the outside of the laminated body 20. The contents ofprocessing performed by the outside three-dimensional distributiongeneration unit 162 are the same as those of the measurement device 10according to the first embodiment or the second embodiment. Then, thedefect specifying unit 166 processes the three-dimensional magneticfield distribution information on the outside of the laminated body 20generated by the outside three-dimensional distribution generation unit162 to specify the depth (z-coordinate) of the defect inside thelaminated body 20 as the information inside the laminated body 20.

More specifically, based on the three-dimensional magnetic fielddistribution information on the outside of the laminated body 20, theboundary condition between the magnetic fields inside and outside thelaminated body 20, and the equation on the magnetic field inside thelaminated body 20, the defect specifying unit 166 specifies the depth ofthe defect inside the laminated body 20. Here, the defect is a defect inwhich a magnetic field of a certain magnitude is localized.

The boundary condition between the magnetic fields inside and outsidethe laminated body 20 is represented by the following expression (10).

$\begin{matrix}{\mspace{79mu}{\left. {\overset{\sim}{H}}_{x} \right|_{inside} = {\left. {\overset{\sim}{H}}_{x} \middle| {}_{outside}\mspace{79mu}{\overset{\sim}{H}}_{y} \right|_{inside} = {\left. {\overset{\sim}{H}}_{y} \middle| {}_{outside}\mspace{79mu}{\overset{\sim}{H}}_{z} \right|_{inside} = {\left. {\overset{\sim}{H}}_{z} \middle| {}_{outside}{\sigma_{a}^{- 1}\left( {{{- {ik}_{y}}{\overset{\sim}{H}}_{z}} - \frac{\partial{\overset{\sim}{H}}_{y}}{\partial z}} \right)} \right|_{outside} = {\left. {\sigma_{c}^{- 1}\left( {{{- {ik}_{y}}{\overset{\sim}{H}}_{z}} - \frac{\partial\overset{\sim}{H_{y}}}{\partial z}} \right)} \middle| {}_{inside}\mspace{79mu}{\sigma_{a}^{- 1}\left( {{{- {ik}_{x}}{\overset{\sim}{H}}_{z}} - \frac{\partial{\overset{\sim}{H}}_{x}}{\partial z}} \right)} \right|_{outside} = {\left. {\sigma_{c}^{- 1}\left( {{{- {ik}_{x}}{\overset{\sim}{H}}_{z}} - \frac{\partial{\overset{\sim}{H}}_{x}}{\partial z}} \right)} \middle| {}_{inside}\mspace{79mu}\frac{\partial{\overset{\sim}{H}}_{z}}{\partial z} \right|_{outside} = \left. \frac{\partial{\overset{\sim}{H}}_{z}}{\partial z} \right|_{inside}}}}}}}} & (10)\end{matrix}$Here,{tilde over (H)} _(x)is the x-component of the complex magnetic field vector,{tilde over (H)} _(y)is the y-component of the complex magnetic field vector,{tilde over (H)} _(z)is the z-component of the complex magnetic field vector, σ_(a) is theelectrical conductivity of outside the laminated body 20, and σ_(c) isthe electrical conductivity of the first layer 210. Here, the firstlayer 210 is the outermost layer on the first plane 201 side of thelaminated body 20.

In addition, an equation relating to the magnetic field inside thelaminated body 20 (time reverse equation) is specifically represented bythe following expression (15). Here, t is the time and Φ is thecomponent in the x-direction or the component in the y-direction of themagnetic field. λ is represented by the following expression (16), μ isthe magnetic permeability inside the laminated body 20, g_(c) is thethickness of the first layer 210, g_(e) is the thickness of the secondlayer 220, σ_(c) is the electrical conductivity of the first layer 210,and σ_(e) is the electrical conductivity of the second layer 220. Themagnetic permeability μ can be regarded as uniform in the outside of thelaminated body 20 and inside the laminated body 20.

$\begin{matrix}{{{- \frac{\partial\;}{\partial t}}\Phi} = {\lambda\;\frac{\partial^{2}}{\partial z^{2}}\Phi}} & (15) \\{\lambda = \frac{g_{c} + g_{e}}{\mu\left( {{\sigma_{c}g_{c}} + {\sigma_{e}g_{e}}} \right)}} & (16)\end{matrix}$

The expression (15) corresponds to the equation (9-9) indicated in thethird embodiment.

Based on the three-dimensional magnetic field distribution informationon the outside of the laminated body 20 and the boundary conditionsbetween the magnetic fields inside and outside the laminated body 20,the defect specifying unit 166 derives (satisfying the expression (15)and specifies the coordinates at which (diverges as the position of thedefect. Specifically, in the boundary condition, the magnetic fieldoutside the laminated body 20 and the differential thereof are derivedfrom the expression (1) of the first embodiment or the expression (12)of the second embodiment. In the present embodiment, here, by settingk_(x)=k_(y)=0, the solution of the expression (15) which is the timereverse equation is obtained. Then, in a case where the z-coordinate atwhich the obtained Φ diverges appears, the coordinate is specified asthe z-coordinate of the defect. Here, the z-coordinate at which Φdiverges specifically refers to the z-coordinate at which the peakappears for Φ on data.

Information which indicates the position (coordinates) of the defect andis output from the defect specifying unit 166 is input to the displayunit 180, for example. Then, on the display unit 180, data fordisplaying the position of the defect in the laminated body 20 isgenerated as an image. Then, it is possible to cause the monitor 390 todisplay the defect position of the laminated body 20 as an image. Inaddition, the display unit 180 may output the coordinates of the defectby displaying characters.

FIGS. 10A to 10D are diagrams illustrating examples of the relationshipbetween z and Φ obtained in the fourth embodiment. This figure is aresult of simulating the relationship between z and Φ as described inthe third embodiment, specifically, it is a graph plotting therelationship between z and Φ using the expression (9-17) In the exampleof this figure, it is assumed that there is a defect at the position ofz₀=10, and Φ_(L)=1 is further assumed. Also, in obtaining this figure,(μσ/2)^(1/2)z was variable-converted to z. Here, the above expression(9-17) makes analysis easier by replacing (1−i) in the expression (9-16)with (α−i). The value of α is −0.1 in FIG. 10A, −0.3 in FIG. 10B, −0.5in FIG. 10C, and −0.7 in FIG. 10D.

As illustrated in this figure, a peak of Φ appeared in the vicinity ofz=10 assuming that a defect exists. Accordingly, it was confirmed thatthe depth of the defect can be specified by the method according to thepresent embodiment. From FIGS. 10A to 10D, it can be understood thatwhen the absolute value of α is increased, resolution is increased. Fromthe simulation results as described above, it can be understood thatwhen any one of the layers in the laminated body 20 has a defect, it ispossible to clarify which layer has a defect by the method of thepresent embodiment.

Next, operations and effects of the present embodiment will bedescribed. In the present embodiment, as in the first embodiment and thesecond embodiment, information on the inside of the laminated body 20such as a battery can be obtained. Furthermore, it provides usefulinformation for non-destructive failure analysis and the like.

In the present embodiment, it is possible to specify the depth of aspecific defect while reducing the load of information processing.Accordingly, in a case where any layer inside the laminated body 20 suchas a battery has a defect, it is possible to specify which layer of thelaminated body 20 has a problem.

Although the embodiments of the present invention have been describedwith reference to the drawings, these are examples of the presentinvention, and various configurations other than the embodimentsdescribed above can be adopted. Further, each of the embodimentsdescribed above can be combined within a range in which the contents donot contradict each other. Also, the mathematical expressions describedabove can be denoted in various ways other than those described aboveand can be appropriately modified and used.

Hereinafter, an example of a reference form will be appended.

1-1. A measurement device including:

a current applying unit which applies a pulse current or a current of aplurality of frequencies to a laminated body having a structure in whicha plurality of layers having different electrical conductivities fromeach other are laminated;

an acquisition unit which acquires in-plane distribution informationincluding at least information indicating distribution of a magneticfield of a first plane outside the laminated body; and

a processing unit which generates three-dimensional magnetic fielddistribution information indicating a three-dimensional distribution ofmagnetic field on an outside of the laminated body based on the in-planedistribution information and generates information on an inside of thelaminated body by processing the three-dimensional magnetic fielddistribution information on the outside of the laminated body,

wherein the in-plane distribution information includes informationindicating response characteristics to a change in current applied bythe current applying unit, and

the three-dimensional magnetic field distribution information on theoutside of the laminated body includes frequency characteristics of amagnetic field.

1-2. In the measurement device described in 1-1,

the laminated body has a structure in which a first layer and a secondlayer which are different from each other in electrical conductivity arelaminated alternately, and

when two directions parallel to the first layer and orthogonal to eachother are defined as the x-direction and the y-direction and a directionperpendicular to the x-direction and the y-direction is the z-direction,the first plane is a plane parallel to the xy-plane.

1-3. In the measurement device described in 1-2,

the in-plane distribution information includes at least magnetic fieldinformation on a magnetic field for each of a plurality of points in thefirst plane, and

the magnetic field information includes information indicating theresponse characteristics of the x-component, the y-component, and thez-component of the magnetic field.

1-4. In the measurement device described in 1-3,

the magnetic field information is represented by a complex magneticfield vector.

1-5. In the measurement device described in 1-3 or 1-4,

when k_(x) is a wave number in the x-direction of the magnetic field,k_(y) is the wave number in the y-direction of the magnetic field, z isa z-coordinate, ω is a frequency, σ is an electrical conductivity, μ isa magnetic permeability, and the following expression (c1) is a complexmagnetic field vector,

the three-dimensional magnetic field distribution information on theoutside of the laminated body is represented by the following expression(c2) and s is represented by the following expression (c3), and

the processing unit obtains the vector a(k_(x), k_(y), ω)) in theexpression (c2) using information indicating the distribution of themagnetic field of the first plane.

$\begin{matrix}{\mspace{79mu}{\overset{\sim}{H}\left( {k_{x},k_{y},z,\omega} \right)}} & ({c1}) \\{\mspace{79mu}{{\overset{\sim}{H}\left( {k_{x},k_{y},z,\omega} \right)} = {{a\left( {k_{x},k_{y},\omega} \right)}e^{sz}}}} & ({c2}) \\{s = {\frac{\sqrt{k_{x}^{2} + k_{y}^{2} + \sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\omega^{2}\sigma^{2}\mu^{2}}}}}{\sqrt{2}} + {i\frac{\sqrt{{- k_{x}^{2}} - k_{y}^{2} + \sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\omega^{2}\sigma^{2}\mu^{2}}}}}{\sqrt{2}}}}} & ({c3})\end{matrix}$1-6. In the measurement device described in any one of 1-2 to 1-4,

the in-plane distribution information includes information indicating adistribution of a magnetic field of the first plane and informationindicating a distribution of a magnetic field of the second plane, and

the second plane is parallel to the first plane and the z-coordinate ofthe second plane is different from that of the first plane.

1-7. In the measurement device described in 1-6,

when k_(x) is a wave number in the x-direction of the magnetic field,k_(y) is the wave number in the y-direction of the magnetic field, z isa z-coordinate, ω is a frequency, σ is an electrical conductivity, μ isa magnetic permeability, and the following expression (c1) is a complexmagnetic field vector,

the three-dimensional magnetic field distribution information on theoutside of the laminated body is represented by the following expression(c4) ands is represented by the following expression (c3), and

the processing unit obtains the vector a(k_(x), k_(y), ω)) and vectorb(k_(x), k_(y), ω)) in the expression (c4) using the informationindicating the distribution of the magnetic field of the first plane andthe information indicating the distribution of the magnetic field of thesecond plane.

$\begin{matrix}{\mspace{79mu}{\overset{\sim}{H}\left( {k_{x},k_{y},z,\omega} \right)}} & ({c1}) \\{\mspace{79mu}{{\overset{\sim}{H}\left( {k_{x},k_{y},z,\omega} \right)} = {{{a\left( {k_{x},k_{y},\omega} \right)}e^{sz}} + {{b\left( {k_{x},k_{y},\omega} \right)}e^{- {sz}}}}}} & ({c4}) \\{s = {\frac{\sqrt{k_{x}^{2} + k_{y}^{2} + \sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\omega^{2}\sigma^{2}\mu^{2}}}}}{\sqrt{2}} + {i\frac{\sqrt{{- k_{x}^{2}} - k_{y}^{2} + \sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\omega^{2}\sigma^{2}\mu^{2}}}}}{\sqrt{2}}}}} & ({c3})\end{matrix}$1-8. In the measurement device described in any one of 1-2 to 1-7,

the processing unit generates the three-dimensional magnetic fielddistribution information on the inside of the laminated body based onthe three-dimensional magnetic field distribution information on theoutside of the laminated body.

1-9. In the measurement device described in 1-8,

the processing unit generates the three-dimensional magnetic fielddistribution on the inside of the laminated body based on thethree-dimensional magnetic field distribution information on the outsideof the laminated body, a boundary condition between the magnetic fieldsinside and outside the laminated body, and a plurality of relationalexpressions to be satisfied by the magnetic field inside the laminatedbody, and

when μ is magnetic permeability, t is time, H_(x) is a component in thex-direction of the magnetic field, H_(y) is a component in they-direction of the magnetic field, H_(z) is a component in thez-direction of the magnetic field, g_(c) is the thickness of the firstlayer, g_(e) is the thickness of the second layer, σ_(c) is theelectrical conductivity of the first layer, and σ_(e) is the electricalconductivity of the second layer,

the plurality of relational expressions are the following expression(c5).

$\begin{matrix}{{{\mu\frac{\partial}{\partial t}H_{x}} = {{\left\{ {{\frac{\left( {{g_{e}\sigma_{c}} + {g_{c}\sigma_{e}}} \right)}{\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}} \right)} + {\frac{\left( {g_{c} + g_{e}} \right)}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)}\frac{\partial^{2}}{\partial z^{2}}}} \right\} H_{x}} + {\frac{g_{e}{g_{c}\left( {\sigma_{c} - \sigma_{e}} \right)}^{2}}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\frac{\partial^{2}}{{\partial x}{\partial z}}H_{z}}}}{{\mu\frac{\partial}{\partial t}H_{y}} = {{\left\{ {{\frac{\left( {{g_{e}\sigma_{c}} + {g_{c}\sigma_{e}}} \right)}{\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}} \right)} + {\frac{\left( {g_{c} + g_{e}} \right)}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)}\frac{\partial^{2}}{\partial z^{2}}}} \right\} H_{y}} + {\frac{g_{e}{g_{c}\left( {\sigma_{c} - \sigma_{e}} \right)}^{2}}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\frac{\partial^{2}}{{\partial y}{\partial z}}H_{z}}}}\mspace{79mu}{{\mu\frac{\partial}{\partial t}H_{z}} = {\frac{\left( {g_{e} + g_{c}} \right)}{\left( {{g_{e}\sigma_{e}} + {g_{c}\sigma_{c}}} \right)}\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}}} \right)H_{z}}}} & ({c5})\end{matrix}$1-10. In the measurement device described in any one of 1-2 to 1-9,

the processing unit specifies the position of a defect inside thelaminated body based on the three-dimensional magnetic fielddistribution information on the outside of the laminated body.

1-11. In the measurement device described in 1-10,

the processing unit specifies a position of the defect inside thelaminated body based on the three-dimensional magnetic fielddistribution information on the outside of the laminated body, theboundary condition between the magnetic fields inside and outside thelaminated body, and the equation on the magnetic field inside thelaminated body.

1-12. In the measurement device described in 1-11,

when μ is magnetic permeability, t is time, H_(x) is a component in thex-direction of the magnetic field, H_(y) is a component in they-direction of the magnetic field, H_(z) is a component in thez-direction of the magnetic field, g_(c) is the thickness of the firstlayer, g_(e) is the thickness of the second layer, σ_(c) is theelectrical conductivity of the first layer, and σ_(e) is the electricalconductivity of the second layer,

the equation is represented by the following expression (c6), and

the processing unit

derives H_(x), H_(y), and H_(z) that satisfy the expression (c6) basedon the three-dimensional magnetic field distribution information on theoutside of the laminated body and a boundary condition between magneticfields inside and outside the laminated body, and

specifies the coordinates at which H_(x), H_(y), and H_(z) diverge asthe position of the defect.

$\begin{matrix}{{{- \mu}\frac{\partial}{\partial t}H_{x}} = {{{\left\{ {{\frac{\left( {{g_{e}\sigma_{c}} + {g_{c}\sigma_{e}}} \right)}{\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}} \right)} + {\frac{\left( {g_{c} + g_{e}} \right)}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)}\frac{\partial^{2}}{\partial z^{2}}}} \right\} H_{x}} + {\frac{g_{e}{g_{c}\left( {\sigma_{c} - \sigma_{e}} \right)}^{2}}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\frac{\partial^{2}}{{\partial x}{\partial z}}H_{z}} - {\mu\;\frac{\partial}{\partial t}H_{z}}} = {{{\left\{ {{\frac{\left( {{g_{e}\sigma_{c}} + {g_{c}\sigma_{e}}} \right)}{\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}} \right)} + {\frac{\left( {g_{c} + g_{e}} \right)}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)}\frac{\partial^{2}}{\partial z^{2}}}} \right\} H_{y}} + {\frac{g_{e}{g_{c}\left( {\sigma_{c} - \sigma_{e}} \right)}^{2}}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\frac{\partial^{2}}{{\partial y}{\partial z}}H_{z}}\mspace{79mu} - {\mu\frac{\partial}{\partial t}H_{z}}} = {\frac{\left( {g_{e} + g_{c}} \right)}{\left( {{g_{e}\sigma_{e}} + {g_{c}\sigma_{c}}} \right)}\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}}} \right)H_{z}}}}} & ({c6})\end{matrix}$2-1. A measurement method which includes:

applying a pulse current or a current of a plurality of frequencies to alaminated body having a structure in which a plurality of layers havingdifferent electrical conductivities from each other are laminated;

acquiring in-plane distribution information including at leastinformation indicating distribution of a magnetic field of a first planeoutside the laminated body; and

generating three-dimensional magnetic field distribution informationindicating a three-dimensional distribution of magnetic field on anoutside of the laminated body based on the in-plane distributioninformation and generating information on an inside of the laminatedbody by processing the three-dimensional magnetic field distributioninformation on the outside of the laminated body,

wherein the in-plane distribution information includes informationindicating response characteristics to a change in current applied tothe laminated body, and

the three-dimensional magnetic field distribution information on theoutside of the laminated body includes frequency characteristics of amagnetic field.

2-2. In the measurement method described in 2-1,

the laminated body has a structure in which a first layer and a secondlayer which are different from each other in electrical conductivity arelaminated alternately, and

when two directions parallel to the first layer and orthogonal to eachother are defined as the x-direction and the y-direction and a directionperpendicular to the x-direction and the y-direction is the z-direction,the first plane is a plane parallel to the xy-plane.

2-3. In the measurement method described in 2-2,

the in-plane distribution information includes magnetic fieldinformation on a magnetic field at least for each of a plurality ofpoints in the first plane, and

the magnetic field information includes information indicating theresponse characteristics of the x-component, the y-component, and thez-component of the magnetic field.

2-4. In the measurement method described in 2-3,

the magnetic field information is represented by a complex magneticfield vector.

2-5. In the measurement method described in 2-3 or 2-4,

when k_(x) is a wave number in the x-direction of the magnetic field,k_(y) is the wave number in the y-direction of the magnetic field, z isa z-coordinate, ω is a frequency, σ is an electrical conductivity, μ isa magnetic permeability, and the following expression (c1) is a complexmagnetic field vector,

the three-dimensional magnetic field distribution information on theoutside of the laminated body is represented by the following expression(c2) and s is represented by the following expression (c3), and

the vector a(k_(x), k_(y), ω)) in the expression (c2) is obtained usinginformation indicating the distribution of the magnetic field of thefirst plane,

$\begin{matrix}{\mspace{79mu}{\overset{\sim}{H}\left( {k_{x},k_{y},z,\omega} \right)}} & ({c1}) \\{\mspace{79mu}{{\overset{\sim}{H}\left( {k_{x},k_{y},z,\omega} \right)} = {{a\left( {k_{x},k_{y},\omega} \right)}e^{sz}}}} & ({c2}) \\{s = {\frac{\sqrt{k_{x}^{2} + k_{y}^{2} + \sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\omega^{2}\sigma^{2}\mu^{2}}}}}{\sqrt{2}} + {i{\frac{\sqrt{{- k_{x}^{2}} - k_{y}^{2} + \sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\omega^{2}\sigma^{2}\mu^{2}}}}}{\sqrt{2}}.}}}} & ({c3})\end{matrix}$2-6. In the measurement method described in any one of 2-2 to 2-4,

the in-plane distribution information includes information indicating adistribution of a magnetic field of the first plane and informationindicating a distribution of a magnetic field of the second plane, and

the second plane is parallel to the first plane and the z-coordinate ofthe second plane is different from that of the first plane.

2-7. In the measurement method described in 2-6,

when k_(x) is a wave number in the x-direction of the magnetic field,k_(y) is the wave number in the y-direction of the magnetic field, z isa z-coordinate, ω is a frequency, σ is an electrical conductivity, μ isa magnetic permeability, and the following expression (c1) is a complexmagnetic field vector,

the three-dimensional magnetic field distribution information on theoutside of the laminated body is represented by the following expression(c4) ands is represented by the following expression (c3), and

the vector a(k_(x), k_(y), ω)) and vector b(k_(x), k_(y), ω)) in theexpression (c4) is obtained using the information indicating thedistribution of the magnetic field of the first plane and theinformation indicating the distribution of the magnetic field of thesecond plane.

$\begin{matrix}{\mspace{79mu}{\overset{\sim}{H}\left( {k_{x},k_{y},z,\omega} \right)}} & ({c1}) \\{\mspace{79mu}{{\overset{\sim}{H}\left( {k_{x},k_{y},z,\omega} \right)} = {{{a\left( {k_{x},k_{y},\omega} \right)}e^{sz}} + {{b\left( {k_{x},k_{y},\omega} \right)}e^{- {sz}}}}}} & ({c4}) \\{s = {\frac{\sqrt{k_{x}^{2} + k_{y}^{2} + \sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\omega^{2}\sigma^{2}\mu^{2}}}}}{\sqrt{2}} + {i\frac{\sqrt{{- k_{x}^{2}} - k_{y}^{2} + \sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\omega^{2}\sigma^{2}\mu^{2}}}}}{\sqrt{2}}}}} & ({c3})\end{matrix}$2-8. In the measurement method described in any one of 2-2 to 2-7,

the three-dimensional magnetic field distribution information on theinside of the laminated body is generated based on the three-dimensionalmagnetic field distribution information on the outside of the laminatedbody.

2-9. In the measurement method described in 2-8,

the three-dimensional magnetic field distribution information on theinside of the laminated body is generated based on the three-dimensionalmagnetic field distribution information on the outside of the laminatedbody, a boundary condition between the magnetic fields inside andoutside the laminated body, and a plurality of relational expressions tobe satisfied by the magnetic field inside the laminated body, and

when μ is magnetic permeability, t is time, H_(x) is a component in thex-direction of the magnetic field, H_(y) is a component in they-direction of the magnetic field, H_(z) is a component in thez-direction of the magnetic field, g_(c) is the thickness of the firstlayer, g_(e) is the thickness of the second layer, σ_(c) is theelectrical conductivity of the first layer, and σ_(e) is the electricalconductivity of the second layer,

the plurality of relational expressions are the following expression

(c5).

$\begin{matrix}{{{\mu\frac{\partial}{\partial t}H_{x}} = {{\left\{ {{\frac{\left( {{g_{e}\sigma_{c}} + {g_{c}\sigma_{e}}} \right)}{\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}} \right)} + {\frac{\left( {g_{c} + g_{e}} \right)}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)}\frac{\partial^{2}}{\partial z^{2}}}} \right\} H_{x}} + {\frac{g_{e}{g_{c}\left( {\sigma_{c} - \sigma_{e}} \right)}^{2}}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\frac{\partial^{2}}{{\partial x}{\partial z}}H_{z}}}}{{\mu\frac{\partial}{\partial t}H_{y}} = {{\left\{ {{\frac{\left( {{g_{e}\sigma_{c}} + {g_{c}\sigma_{e}}} \right)}{\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}} \right)} + {\frac{\left( {g_{c} + g_{e}} \right)}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)}\frac{\partial^{2}}{\partial z^{2}}}} \right\} H_{y}} + {\frac{g_{e}{g_{c}\left( {\sigma_{c} - \sigma_{e}} \right)}^{2}}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\frac{\partial^{2}}{{\partial y}{\partial z}}H_{z}}}}\mspace{79mu}{{\mu\frac{\partial}{\partial t}H_{z}} = {\frac{\left( {g_{e} + g_{c}} \right)}{\left( {{g_{e}\sigma_{e}} + {g_{c}\sigma_{c}}} \right)}\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}}} \right)H_{z}}}} & ({c5})\end{matrix}$2-10. In the measurement method described in any one of 2-2 to 2-9,

the position of the defect inside the laminated body is specified basedon the three-dimensional magnetic field distribution information on theoutside of the laminated body.

2-11. In the measurement method described in 2-10,

the position of the defect inside the laminated body is specified basedon the three-dimensional magnetic field distribution information on theoutside of the laminated body, the boundary condition between themagnetic fields inside and outside the laminated body, and the equationon the magnetic field inside the laminated body.

2-12. In the measurement method described in 2-11,

when μ is magnetic permeability, t is time, H_(x) is a component in thex-direction of the magnetic field, H_(y) is a component in they-direction of the magnetic field, H_(z) is a component in thez-direction of the magnetic field, g_(c) is the thickness of the firstlayer, g_(e) is the thickness of the second layer, σ_(c) is theelectrical conductivity of the first layer, and σ_(e) is the electricalconductivity of the second layer,

the equation is represented by the following expression (c6), H_(x),H_(y), and H_(z) that satisfy the expression (c6) are derived based onthe three-dimensional magnetic field distribution information on theoutside of the laminated body and a boundary condition between amagnetic field inside and outside of the laminated body, and

the coordinates at which H_(x), H_(y), and H_(z) diverge are specifiedas the position of the defect.

$\begin{matrix}{{{- \mu}\frac{\partial}{\partial t}H_{x}} = {{{\left\{ {{\frac{\left( {{g_{e}\sigma_{c}} + {g_{c}\sigma_{e}}} \right)}{\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}} \right)} + {\frac{\left( {g_{c} + g_{e}} \right)}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)}\frac{\partial^{2}}{\partial z^{2}}}} \right\} H_{x}} + {\frac{g_{e}{g_{c}\left( {\sigma_{c} - \sigma_{e}} \right)}^{2}}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\frac{\partial^{2}}{{\partial x}{\partial z}}H_{z}} - {\mu\;\frac{\partial}{\partial t}H_{z}}} = {{{\left\{ {{\frac{\left( {{g_{e}\sigma_{c}} + {g_{c}\sigma_{e}}} \right)}{\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}} \right)} + {\frac{\left( {g_{c} + g_{e}} \right)}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)}\frac{\partial^{2}}{\partial z^{2}}}} \right\} H_{y}} + {\frac{g_{e}{g_{c}\left( {\sigma_{c} - \sigma_{e}} \right)}^{2}}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\frac{\partial^{2}}{{\partial y}{\partial z}}H_{z}}\mspace{79mu} - {\mu\frac{\partial}{\partial t}H_{z}}} = {\frac{\left( {g_{e} + g_{c}} \right)}{\left( {{g_{e}\sigma_{e}} + {g_{c}\sigma_{c}}} \right)}\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}}} \right)H_{z}}}}} & ({c6})\end{matrix}$

This application is based upon and claims the benefit of priority of theprior Japanese Patent Application No. 2016-090128, filed on Apr. 28,2016, the entire contents of which are incorporated herein by reference.

The invention claimed is:
 1. A measurement device comprising: a currentgenerator which applies a current to a laminated body having a structurein which a plurality of layers having different electricalconductivities from each other are laminated, the current being a pulsecurrent or a current of a plurality of frequencies and is applied to afirst electrode to cause the current to flow through the plurality oflayers to a second electrode; a magnetic sensor which acquires in-planedistribution information including at least information indicatingdistribution of a magnetic field at the plurality of frequencies in afirst plane outside the laminated body; and an information processingdevice which generates three-dimensional magnetic field distributioninformation indicating a three-dimensional distribution of magneticfield on an outside of the laminated body based on the in-planedistribution information and generates information on an inside of thelaminated body by processing the three-dimensional magnetic fielddistribution information on the outside of the laminated body, whereinthe in-plane distribution information includes frequency dependentcomplex data which is information indicating response characteristics toa change in the current at the plurality of frequencies applied by thecurrent generator, the three-dimensional magnetic field distributioninformation on the outside of the laminated body includes frequencycharacteristics of the magnetic field at the plurality of frequencies,the information processing device specifies a boundary condition betweenthe magnetic fields inside and outside the laminated body, and anequation relating to the magnetic field inside the laminated body, andthe equation is an averaged and continuous diffusion type partialdifferential equation of the laminated body, wherein the informationprocessing device, based on the boundary condition between the magneticfields inside and outside the laminated body and the frequencycharacteristics of the magnetic field of the three-dimensional magneticfield distribution information at the plurality of frequencies, derivesthree-dimensional magnetic field vectors that satisfy the equation, anddetermines that a position of a defect inside the laminated body islocated at a three-dimensional coordinate where the three-dimensionalmagnetic field vectors diverge, the laminated body is a battery, theplurality of layers include positive electrodes of the battery andnegative electrodes of the battery, at least one of the positiveelectrodes of the battery is electrically connected to the firstelectrode, and at least one of the negative electrodes of the battery iselectrically connected to the second electrode.
 2. The measurementdevice according to claim 1, wherein the laminated body has a structurein which a first layer and a second layer which are different from eachother in electrical conductivity are laminated alternately, and when twodirections parallel to the first layer and orthogonal to each other arean x-direction and a y-direction and a direction perpendicular to thex-direction and the y-direction is a z-direction, the first plane is aplane parallel to an xy-plane.
 3. The measurement device according toclaim 2, wherein the in-plane distribution information includes magneticfield information on a magnetic field at least for each of a pluralityof points in the first plane and at the plurality of frequencies, andthe magnetic field information includes information indicating theresponse characteristics of an x-component, a y-component, and az-component of the magnetic field at the plurality of frequencies in thefirst plane outside the laminated body.
 4. The measurement deviceaccording to claim 3, wherein the magnetic field information isrepresented by a complex magnetic field vector.
 5. The measurementdevice according to claim 3, wherein the three-dimensional magneticfield distribution information on the outside of the laminated body atthe plurality of frequencies is represented by the following expressions(c1) and (c2), and is represented by the following expression (c3), whenk_(x) is a wave number in the x-direction of the magnetic field, k_(y)is a wave number in the y-direction of the magnetic field, z is az-coordinate, ω is a frequency, σ is electrical conductivity, μ ismagnetic permeability, and the information processing device obtains thevector a(k_(x), k_(y), ω) in the expression (c2) using the informationindicating the distribution of the magnetic field at the plurality offrequencies of the first plane, $\begin{matrix}{\overset{˜}{H}\left( {k_{x},k_{y},z,\omega} \right)} & ({c1}) \\{{\overset{\sim}{H}\left( {k_{x},k_{y},z,\omega} \right)} = {{a\left( {k_{x},k_{y},\omega} \right)}e^{sz}}} & ({c2}) \\{s = {\frac{\sqrt{k_{x}^{2} + k_{y}^{2} + \sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\omega^{2}\sigma^{2}\mu^{2}}}}}{\sqrt{2}} + {i{\frac{\sqrt{{- k_{x}^{2}} - k_{y}^{2} + \sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\omega^{2}\sigma^{2}\mu^{2}}}}}{\sqrt{2}}.}}}} & ({c3})\end{matrix}$
 6. The measurement device according to claim 2, whereinthe in-plane distribution information includes the informationindicating distribution of a magnetic field of the first plane and atthe plurality of frequencies, and information indicating distribution ofa magnetic field of a second plane and at the plurality of frequencies,and the second plane is parallel to the first plane, and thez-coordinate of the second plane is different from that of the firstplane.
 7. The measurement device according to claim 6, wherein thethree-dimensional magnetic field distribution information on the outsideof the laminated body at the plurality of frequencies is represented bythe following expressions (c1) and (c4), and s is represented by thefollowing expression (c3), wherein when k_(x) is a wave number in thex-direction of the magnetic field, k_(y) is a wave number in they-direction of the magnetic field, z is a z-coordinate, ω is afrequency, σ is electrical conductivity, μ is magnetic permeability, andthe information processing device obtains the vector a(k_(x), k_(y), ω)and vector b(k_(x), k_(y), ω) in the expression (c4) using theinformation indicating the distribution of the magnetic field of thefirst plane at the plurality of frequencies and the informationindicating the distribution of the magnetic field of the second plane atthe plurality of frequencies, $\begin{matrix}{\overset{˜}{H}\left( {k_{x},k_{y},z,\omega} \right)} & ({c1}) \\{{\overset{˜}{H}\left( {k_{x},k_{y},z,\omega} \right)} = {{{a\left( {k_{x},k_{y},\omega} \right)}e^{sz}} + {{b\left( {k_{x},k_{y},\omega} \right)}e^{{- s}z}}}} & ({c4}) \\{s = {\frac{\sqrt{k_{x}^{2} + k_{y}^{2} + \sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\omega^{2}\sigma^{2}\mu^{2}}}}}{\sqrt{2}} + {i{\frac{\sqrt{{- k_{x}^{2}} - k_{y}^{2} + \sqrt{\left( {k_{x}^{2} + k_{y}^{2}} \right)^{2} + {\omega^{2}\sigma^{2}\mu^{2}}}}}{\sqrt{2}}.}}}} & ({c3})\end{matrix}$
 8. The measurement device according to claim 2, whereinthe information processing device generates the three-dimensionalmagnetic field distribution information on the inside of the laminatedbody based on the three-dimensional magnetic field distributioninformation on the outside of the laminated body at the plurality offrequencies.
 9. The measurement device according to claim 8, wherein theinformation processing device generates the three-dimensional magneticfield distribution information on the inside of the laminated body basedon the three-dimensional magnetic field distribution information on theoutside of the laminated body at the plurality of frequencies, theboundary condition between the magnetic fields inside and outside thelaminated body at the plurality of frequencies, and a plurality ofrelational expressions to be satisfied by the magnetic field inside thelaminated body at the plurality of frequencies, and when μ is magneticpermeability, t is time, H_(x) is a component in the x-direction of themagnetic field, H_(y) is a component in the y-direction of the magneticfield, H_(z) is a component in the z-direction of the magnetic field,g_(c) is a thickness of the first layer, g_(e) is a thickness of thesecond layer, σ_(c) is the electrical conductivity of the first layer,and σ_(e) is the electrical conductivity of the second layer, theplurality of relational expressions are the following expression (c5),$\begin{matrix}\begin{matrix}{{\mu\frac{\partial}{\partial t}H_{x}} = {\left\{ {{\frac{\left( {{g_{e}\sigma_{c}} + {g_{c}\sigma_{e}}} \right)}{\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\left( {\frac{\partial^{2}}{\partial x^{2}}{+ \frac{\partial^{2}}{\partial y^{2}}}} \right)} + {\frac{\left( {g_{c} + g_{e}} \right)}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)}\frac{\partial^{2}}{\partial z^{2}}}} \right\} H_{x}}} \\{{+ \frac{g_{e}{g_{c}\left( {\sigma_{c} - \sigma_{e}} \right)}^{2}}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}}\frac{\partial^{2}}{{\partial x}{\partial z}}H_{z}} \\{{\mu\frac{\partial}{\partial t}H_{y}} = {\left\{ {{\frac{\left( {{g_{e}\sigma_{c}} + {g_{c}\sigma_{e}}} \right)}{\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\left( {\frac{\partial^{2}}{\partial x^{2}}{+ \frac{\partial^{2}}{\partial y^{2}}}} \right)} + {\frac{\left( {g_{c} + g_{e}} \right)}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)}\frac{\partial^{2}}{\partial z^{2}}}} \right\} H_{y}}} \\{{+ \frac{g_{e}{g_{c}\left( {\sigma_{c} - \sigma_{e}} \right)}^{2}}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}}\frac{\partial^{2}}{{\partial y}{\partial z}}H_{z}} \\{{\mu\frac{\partial}{\partial t}H_{z}} = {\frac{\left( {g_{e} + g_{c}} \right)}{\left( {{g_{e}\sigma_{e}} + {g_{c}\sigma_{c}}} \right)}\left( {\frac{\partial^{2}}{\partial x^{2}}{+ {\frac{\partial^{2}}{\partial y^{2}}{+ \frac{\partial^{2}}{\partial z^{2}}}}}} \right){H_{z}.}}}\end{matrix} & ({c5})\end{matrix}$
 10. The measurement device according to claim 2, whereinthe information processing device generates the three-dimensionalmagnetic field distribution information on the inside of the laminatedbody based on the three-dimensional magnetic field distributioninformation on the outside of the laminated body at the plurality offrequencies, the boundary condition between the magnetic fields insideand outside the laminated body at the plurality of frequencies, and aplurality of relational expressions to be satisfied by the magneticfield inside the laminated body at the plurality of frequencies, andwherein when μ is magnetic permeability, t is time, H_(x) is a componentin the x-direction of the magnetic field, H_(y) is a component in they-direction of the magnetic field, H_(z) is a component in thez-direction of the magnetic field, g_(c) is the thickness of the firstlayer, g_(e) is the thickness of the second layer, σ_(c) is theelectrical conductivity of the first layer, and σ_(e) is the electricalconductivity of the second layer, the plurality of relationalexpressions are the following expression (c6), $\begin{matrix}\begin{matrix}{{{- \mu}\frac{\partial}{\partial t}H_{x}} = {\left\{ {{\frac{\left( {{g_{e}\sigma_{c}} + {g_{c}\sigma_{e}}} \right)}{\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\left( {\frac{\partial^{2}}{\partial x^{2}}{+ \frac{\partial^{2}}{\partial y^{2}}}} \right)} + {\frac{\left( {g_{c} + g_{e}} \right)}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)}\frac{\partial^{2}}{\partial z^{2}}}} \right\} H_{x}}} \\{{+ \frac{g_{e}{g_{c}\left( {\sigma_{c} - \sigma_{e}} \right)}^{2}}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}}\frac{\partial^{2}}{{\partial x}{\partial z}}H_{z}} \\{{{- \mu}\frac{\partial}{\partial t}H_{y}} = {\left\{ {{\frac{\left( {{g_{e}\sigma_{c}} + {g_{c}\sigma_{e}}} \right)}{\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}\left( {\frac{\partial^{2}}{\partial x^{2}}{+ \frac{\partial^{2}}{\partial y^{2}}}} \right)} + {\frac{\left( {g_{c} + g_{e}} \right)}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)}\frac{\partial^{2}}{\partial z^{2}}}} \right\} H_{y}}} \\{{+ \frac{g_{e}{g_{c}\left( {\sigma_{c} - \sigma_{e}} \right)}^{2}}{\left( {{g_{c}\sigma_{c}} + {g_{e}\sigma_{e}}} \right)\left( {g_{e} + g_{c}} \right)\sigma_{c}\sigma_{e}}}\frac{\partial^{2}}{{\partial y}{\partial z}}H_{z}} \\{{{- \mu}\frac{\partial}{\partial t}H_{z}} = {\frac{\left( {g_{e} + g_{c}} \right)}{\left( {{g_{e}\sigma_{e}} + {g_{c}\sigma_{c}}} \right)}\left( {\frac{\partial^{2}}{\partial x^{2}}{+ {\frac{\partial^{2}}{\partial y^{2}}{+ \frac{\partial^{2}}{\partial z^{2}}}}}} \right){H_{z}.}}}\end{matrix} & ({c6})\end{matrix}$
 11. A measurement method comprising: applying a current toa laminated body having a structure in which a plurality of layershaving different electrical conductivities from each other arelaminated, where the applying the current comprises a pulse current or acurrent of a plurality of frequencies and is applied to a firstelectrode to cause the current to flow through the plurality of layersto a second electrode, and wherein the laminated body comprises abattery and the plurality of layers include positive electrodes of thebattery and negative electrodes of the battery; acquiring in-planedistribution information including at least information indicatingdistribution of a magnetic field at the plurality of frequencies in afirst plane outside the laminated body; and generating three-dimensionalmagnetic field distribution information indicating a three-dimensionaldistribution of magnetic field on an outside of the laminated body basedon the in-plane distribution information and generating information onan inside of the laminated body by processing the three-dimensionalmagnetic field distribution information on the outside of the laminatedbody, wherein the in-plane distribution information includes frequencydependent complex data which is information indicating responsecharacteristics to a change in current at the plurality of frequenciesapplied to the laminated body, the three-dimensional magnetic fielddistribution information on the outside of the laminated body includesfrequency characteristics of the magnetic field at the plurality offrequencies, in the generating of the information on the inside of thelaminated body, a boundary condition between the magnetic fields insideand outside the laminated body, and an equation relating to the magneticfield inside the laminated body are specified, the equation is anaveraged and continuous diffusion type partial differential equation ofthe laminated body; and based on the boundary condition between themagnetic fields inside and outside the laminated body and the frequencycharacteristics of the magnetic field at the plurality of frequencies ofthe three-dimensional magnetic field distribution information, derivingthree-dimensional magnetic field vectors that satisfy the equation, anddetermining that a position of a defect inside the laminated body islocated at a three-dimensional coordinate where the three-dimensionalmagnetic field vectors diverge, electrically connecting at least one ofthe positive electrodes of the battery to the first electrode, andelectrically connecting at least one of the negative electrodes of thebattery to the second electrode.